Author: groesbecks

High School Algebra: Systems of Equations

groesbecks CCSS-Math Learning Progressions, HS Algebra, ML Expressions & Equations

system-linear-types

This learning progression will be applied in an Algebra I classroom where most students are in the 10th grade. This classroom has access to CORD’s Algebra 1: Learning in Context digital textbook. The common core state standards aligned with this learning progression are 8.EE.C.8, 8.EE.C.8A, 8.EE.C.8B, and HSA.REI.C.6. This learning progression is also aligned to the following mathematical practices: MP1- Make sense of problems and persevere in solving them, MP4- Model with mathematics, MP5-Use appropriate tools strategically, and MP6- Attend to precision.

 

Day 1 Instructional Task 1: In order to introduce the students to the concept of systems of linear equations, the teacher will start with by having two students volunteer to act out an example. The teacher will have them stand on opposite sides of the classroom and then walk towards each other. While they are doing that, the rest of the class will be told to watch them closely. The class will notice that at some point the students will cross paths before they continue on their respective routes. The teacher will lead a short discussion relating this walking exercise to systems of linear equations, specifically those whose graphs intersect (which the students will later learn are called consistent and independent systems). The teacher can then take this activity a step further by showing the students one way systems of linear equations can be used in the real world, in other words a way to create a system of linear equations model.

Benchmark Assessment Day 1: This will be in the form of an exit task. Students will be given the following question: “In your own words, explain what a common solution of a system of linear equations represents graphically and explain why it is significant algebraically.” This assessment fits in with a Standards Based Grading system and is aligned with standard 8.EE.C.8.A.

 

Learning Progression for edTPA

edTPA Lesson Plan

High School Algebra 1: Functions… sounds like a party!

groesbecks CCSS-Math Learning Progressions, Common Core State Standards Mathematics, HS Algebra, HS Functions

function-parts

 

Before jumping into functions, one should first look at relations (sets of ordered pairs) and patterns in order to gain and build the meaningful knowledge required when working with functions and their domains and ranges. This learning progression does just that. This learning progression will be applied in an Algebra I classroom where most students are in the 10th grade. This classroom has access to CORD’s Algebra 1: Learning in Context digital textbook.

The common core state standards aligned with this learning progression are HSF.IF.A.1 and HSF.IF.A.2. This learning progression is also aligned to the following mathematical practices: MP1- Make sense of problems and persevere in solving them, MP6- Attend to precision, and MP7- Look for and make use of structure.

This learning progression incorporates the mathematical best practice of using a variety of continuous assessments that are designed to not only measure student understanding but also teacher effectiveness. From these assessments, the teacher will be able to tell which lessons resonated with the students and which fell flat and need to be retaught. Using assessment in this way reinforces that assessment is more than just a way for teachers to grade students; it is an integral part of teaching.

Algebra 1 Functions- Learning Progression

A-CED: Rearranging Formulas- IM Assessment

groesbecks Assessment of CCSS-Math, Common Core State Standards Mathematics, HS Algebra

rearrangements

This IM Assessment is aligned to the following Common Core State Standard:

CCSS.Math.Content.HSA.CED.A.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

image

For the complete assessment, commentary, and solutions please see the attached pdf document below.

Math 325-IM Assessment on rearranging formulas

Transformations in Architecture- CCSS.Math.Content.8.G.A.4

groesbecks HS Geometry, Picture Problems

40190988

(Picture retrieved from: http://commondatastorage.googleapis.com/static.panoramio.com/photos/original/40190988.jpg )

Above is Maqbaratoshoara or “The Mausoleum of Poets” which is located in Iran. In the architecture of this building, you can find similar shapes that result from various transformations. It would be best to present this picture and the following questions to students after they have been introduced to reflections and dilations.

Find the similar shapes in this building:

  • How where these similar shapes obtained?
  • How many reflections do you see?
  • Where is/are the line(s) of symmetry?
  • How many dilations do you see?
  • Where is/are the center(s) of dilation?
  • Do you see any other transformation in this picture?

 

This picture problem meets the following Common Core State Standard:

CCSS.Math.Content.8.G.A.4
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Hey! Wait Up!- CCSS.MATH.CONTENT.8.EE.C.8

groesbecks HS Algebra, HS Modeling, ML Expressions & Equations, ML Modeling, Technology Teaching Issues, Uncategorized

Oracles-Chasing-DreamImagine you are walking to class and you see your friend up ahead so you run to catch up with him. Can you model this situation mathematically? Of course you can! Assuming that you and your friend are moving at constant speeds, you can set up a system of linear equations. You can model this graphically by plotting distance versus time. If you were to plot the lines on the same set of axes, the point where the two lines cross would represent the physical location where you passed your friend.

VernierCBR2

It is possible to create models for situations like this in the classroom using Vernier CBR2 Motion Detectors.

In the Hey! Wait Up! activity, students will work in groups of four: 2 people will be the walkers, 1 person will start the motion detector and operate the stopwatch, and 1 person will mark the point where the walkers intersect. Each group will work together to collect and analyze motion data in order to determine the solution to a linear system of equations using a graphing calculator. They will then check this solution by creating a system of linear equations and solving it by hand. This activity is best done after the students have been introduced  to systems of linear equations and how to solve them.

To complete the The Hey! Wait Up! activity each group will need 2 Vernier CBR2 Motion Detectors (with appropriate cables), a TI83 or TI84 calculator, a meter stick, and a stopwatch.

 

This lesson aligns with the following Common Core State Standards:

CCSS.MATH.CONTENT.8.EE.C.8– Analyze and solve pairs of simultaneous linear equations.

Screen Shot 2015-10-29 at 12.29.58 PMStudents will be using the calculator results to find the coordinates of two points for both lines which they will then use to find the slopes of each line. Students will also use the calculator results to find the y-intercept of each line. Using this information, students will be able to create equations for the lines that they will then use to make a system of linear equations and solve.

Screen Shot 2015-10-29 at 12.37.44 PM

CCSS.MATH.CONTENT.8.EE.C.8.A– Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

At the end of this activity, students will be able to connect that the point where they crossed when walking is the point where the two lines intersect and that this coordinate point is the solution to the system of linear equations.

 

To conclude this activity, students will be given an exit slip that asks:

  • In your own words, what is a system of linear equations?
  • Give me an example of a real world situation (that does not involve walking/ running) that you could model using a system of linear equations.

From this you can determine and gauge your students’ conceptual understanding as well as see if they can apply the idea of using systems of linear equations to model other everyday experiences.

 

A benefit of using Vernier Motion Detectors when teaching the concept of systems of linear equations is that it makes the concept personal for the students. It helps bring mathematics out of the classroom and into the real world. Rather than seeing the systems of linear equations as just lines on the graph, they can see that in this situation those lines represent something, namely the students’ walks. This activity also gets the students involved in their own learning by having them get up and move around when they are doing the walking activities and collecting the data. This makes the data concrete for them rather than just some numbers, equations, and coordinates they have to work with in order to find a solution.

Vernier Logo

A Vernier CBR2 Motion Detector costs $99. To buy or to find out more about the Vernier CBR2 Motion Detector visit http://www.vernier.com/products/sensors/motion-detectors/cbr2/

To find out more about Vernier Software & Technology and explore their other products visit http://www.vernier.com/

 

Hey! Wait Up! worksheet: Hey! Wait Up!-Systems of Linear Equations with Vernier Motion Detectors

Oh Darn! I left my tape measure at home… but it’s okay I have my math tools. CCSS.Math.Content.HSG.SRT.C.8

groesbecks HS Geometry, HS Modeling, Technology Teaching Issues, Uncategorized

Modeling_picEver wonder how long your shadow was or how tall a building was? These questions are answerable without having to break out a measuring tape. You can answer these questions using trigonometric ratios and the Pythagorean Theorem.

In this lesson, students will be given real world word problems involving right triangles for which they will create mathematical models and then use trigonometric ratios and the Pythagorean Theorem to find the unknown sides and angles of the given triangle. Students will create these models by hand as well as by using GeoGebra.

GeoGebra is a free program that you can get on your computer and your tablet. GeoGebra can be used to teach and learn geometry, algebra, statistics, and calculus. It is fairly simple to use and can be used by people of all ages. In this lesson, students will be using the geometry side of GeoGebra. Students will be creating a model similar to the picture above to find the length of the new sidewalk between the Science building and the Café.

Screen Shot 2015-10-22 at 5.46.37 PM

This lesson meets the High School Geometry Common Core State Standard: CCSS.Math.Content.HSG.SRT.C.8- Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

Modeling using Trig Ratios and Pythagorean Theorem Lesson Plan

Modeling using Trig Ratios and Pythagorean Theorem Worksheet

 

CCSS.MATH.CONTENT.HSS.ID.A.1: Using Alcula to create a Box and Whisker Plots

groesbecks HS Modeling, HS Statistics & Probability, Technology Teaching Issues, Uncategorized

Screen Shot 2015-10-20 at 4.10.50 PM

Attached is an article about Alcula, a website that can be used to create box and whisker plots. This website is an easy, accessible tool that empowers students to understand box and whisker plots on a deeper level. Students can check their mathematical understanding by using Alcula to verify their answers for the 5 number summary of a given data set and the box and whisker plots they have created by hand.

Teachers can use this website during an entry task and have students determine the 5 number summary for a given box and whisker plot. Teachers can have their students use this website individually or in small groups. Teachers can also use Alcula to give the students a visual representation of test results and grades. Alcula can be used on a computer, smart phone, or tablet that has internet access.

Using Alcula in the Classroom Article