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.

SBAC Practice Activities: The Park

This assignment is modeled after the first part of an 8th grade performance task released by SBAC called "The Park".  It is intended for students in grades 9-11.  We chose to focus on lower level content to give as many students as possible access to the problem while focusing on using technology to show work and explain thinking.  Our modified Park task asks students to make use of a grant to design a park given certain restrictions.  Students are finding the lowest bid to place benches, lamp posts, and bathrooms in the park.  A copy of the assignment can be viewed here, and screenshots of the assignment will be shown below.

Diagram 1:  Students receive the page below via Google Drive.  All of the information that they need to solve the problem can be found at the various links (shown in yellow, green, blue, and purple), and students will give all final answers/work on this page.  As with SBAC performance tasks, we spend some time together making sure that students understand the relevant vocabulary.  We also walked students through the task and fielded any questions that came up before the work period began.

Diagram 2:  When students open the "Introduction and Park Specifications" page from the yellow box, they see the park below, along with a description of the grant requirements.  The requirements are to place benches all along the three paths in the middle of the park, beginning by placing three benches where the paths meet.  Next to each bench we place a lamppost, and then also we add two buildings that contain restrooms.

Diagram 3:  Students will open the "Work Bids" page from the green box, and view the three bids below.  

Diagram 4:  Students will open the "Interactive Diagram" page.  We give students the choice to either use the interactive diagram to add benches/lampposts, or to draw on a piece of paper.  The important part is that student work will only be scored from the original Google Doc.  This is like the SBAC test in that paper is a tool for answering questions, but all final answers will be inputted on the computer.

The last link on the Google Doc opens up the SBAC calculator.  This is the calculator that students will be using on the SBAC test, so we want to make sure students get a chance to practice using the calculator before the actual exam.

While students work on the problem, the teacher is circulating to coach students on best practices for answering the questions.  We emphasize three key areas, and encourage students to think about each area as earning them 1 point towards a final score.  These key areas are:

• 1 point for showing your work
• 1 point for explaining your thinking 
• 1 point for giving your final answer with units

SBAC Equation Editor and Some Notes on Formulas

Below are a few screenshots from the SBAC Equation Editor tutorial.  After the SBAC Interim tests, students may or may not need additional practice inputting answers that include formulas/variables. If practice is needed, please direct students to the tutorial, where they can get immediate feedback on whether or not they are entering answers correctly.

Students can pick from any of the tutorials below.  For these screenshots I am using the Exponents tutorial.

This tutorial walks me through directions on how to enter three different expressions.  The first set of directions are below.

This particular tutorial walked me through how to enter the following three expressions.

When finished, press the button at the bottom left to see how you did.

Score results are below. (I changed expression 3 from what you see in the image above so that we could see what feedback is given for incorrect answers.)

I also learned that standard shortcuts like "^" for raising to a power and "sqrt" for square root still work.

A couple of additional observations:  

• I haven't seen the use of subscripts yet in any of the released problems.  It appears that for sequences students will write f(n)=3n+2 or s(n)=4x-7 (etc.).  Also for recursive rules we will write f(1)=10, f(n)=f(n-1)+3 for n>1 for the sequence with first term of 10 that increases by 3 for each term.
• I have not seen any formulas given for volume.  I have also not seen problems where students are asked to find the volume of cones or pyramids. So far I have only seen rectangular prisms, cylinders, and spheres.

Anything other observations we can note here?  Please email Shelley or add to the comments.

Suggestions for Future SBAC Blogposts???

You can view the results in a spreadsheet at this link.

Desmos Circle Challenges and Explorations

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.

Where's the Geometry: SBAC and claims 2-3

As mentioned the other day, there are very few Geometry problems that are included in the Claim 1 section of the SBAC blueprint.  In fact, there will only be 2 Geometry problems tested at the Claim 1 (concepts and procedures) level of the test.  They will both be from Target O, which is "Define trigonometric ratios and solve problems involving right triangles.  SBAC has provided us with many sample problems from this target.  You can see two of the Claim 1 problems involving trig ratios and right triangles below.

Note:  I didn't see a calculator involving inverse trig functions, nor a table of trig values, so I'm not sure how they want students to solve this.

You might be asking yourself, "what about the other Geometry concepts?"  It turns out that most of the Geometry standards/targets are tested at the Claim 2, 3 or 4 level.  The rest of today's blogpost will be about those Geometry problems tested at the Claim 2 or 3 level.

Remember, Claim 2 is about problem solving, and Claim 3 is about Communicating Reasoning.

Other problems from trig/right triangles are also included in Claims 2 and 3.

The question above asks students to solve for h in terms of b.

What about the other Geometry standards? How do they appear on this test?  The answer is that they appear as Claim 2, 3, or 4 questions.  Students will be expected to display problem solving skills and communicate reasoning as they pertain the Geometry.  They also will be expected to apply knowledge of Geometry to modeling problems (Claim 4), which we will look at in a few weeks.  Below are some additional sample problems from Geometry.

 Parallel Lines:

Transformations:

Congruent Triangles:

Circles:

Area:

Volume

There will be additional problems on the Performance Tasks, which count as Claim 4.  Most of the Geometry I have seen so far on the performance tasks includes trig/right triangles as well as area and volume.  More to come on this in a few weeks.  Click here for a link to a word document that contains all the problems from this post.

SBAC Math Rubrics for Scoring

All SBAC problems are worth 1, 2, 3 or 4 points.  For performance tasks, the problem may be broken into parts, with each being worth 1-4 points.  Below is a generic 4-point rubric provided by Smarter Balanced.  You can view the generic rubrics for 3, 2, and 1 point problems here.

Many of the released items from Smarter Balanced include a rubric, modeled after the generic rubric, which provides additional details about how the problem will be scored.  Below is Sample Item 43028, called "ebooks", from this website.  The rubric for "ebooks" is below the problem.

Rubric for "ebooks".

The performance task problems are longer, and so the scoring is broken down into different parts for the problem.  Below is the introduction to the performance task called "Thermometer Crickets".  In this task, students are asked to find a line of best fit for a set of data, and compare this model to a classic model for chirps-per-minute to temperature called Dolbear's Law.  The rubric for "Thermometer Crickets" is posted below the problem.

The rubrics for the SBAC Practice Test can be found here.

SBAC: The 4 Claims and how they are tested

Today's post will elaborate on the SBAC Blueprint that was sent out yesterday.

Details about Claim 1:

There is only one target for Geometry and one for Statistics/Probability under Claim 1.  From the chart yesterday we learned that we will see two problems from each of these strands.  The other strands include Numbers, Algebra, and Functions, and together will include 18 problems.  Remember, Claim 1 problems will appear on part I of the SBAC test.  Part II of the SBAC test is the performance task.  More information about the Geometry and Statistic/Probability strands can be seen below.

A few weeks back I sent out this document, which includes all of the released questions that I could find related to Statistics and Probability.  For Geometry I have seen some released problems about equations of circles as well as the triangle congruence theorems.  I am still learning how that fits into the blueprint.  The answer to this question, plus more on the Claims to come later this week.

This information was taken from this link.

SBAC Blueprint: How many problems and what type

Below are two diagrams that show the types of questions and main content standards that will be covered on the SBAC.

Diagram 1:  CAT stands for computer adaptive testing, and includes the problems that can be scored by the computer.  PT stands for performance task, and is generally an extended problem that students will need to explain their thinking for.  I am still learning about the details, but it looks like we can expect 33 shorter problems and 6 longer problems.  The break the test into two parts.  The first part is most of the shorter problems.  The second part is called the performance task and is an extended, multi-part question that students will answer in one sitting.

Diagram 2:  In the table below you can see the breakdown of content that we can expect to see on the test.  The majority of problems will be taken from the priority clusters, though problems from the supporting clusters will be included.  The third diagram is a continuation of diagram 2 (forgot to include this when post first published).

This information is from the SBAC Blueprint.  More on this in the next few days.  Please let Shelley know if there are specific questions to answer or suggestions for what to cover over the next few weeks.

Pictures of the circle activity

 The glue 20 of these together, small triangles on the inside, large triangles on the outside to make an icosahedron.