High School: Algebra Seeing Structure in Equations And High School: Functions Interpreting Functions

 

This learning progression will take place in a 9th/10th grade Algebra class at Ellensburg High School. These lessons will align with the mathematics content standards 8.EE.A.1., HSA.SSE.B.3.C, HSF.IF.C.7.E, and HSF.IF.C.8.B, the textbook that will be used is McDougal Littell Algebra 1 2004. The learning progression will also be aligned with these mathematical practices: MP4 Model with mathematics and MP6 Attend to precision.

Students have spent the year learning about and solving equations and expressions using the order of operations. They have used their skills and knowledge to solve linear functions and inequalities as well as find representations for data from charts and graphs. For this learning progression they will be using their skills in order of operations as well as adding new exponent properties to solve and evaluate exponential problems and eventually be able to model exponential growth and decay with their exponential skills.

These lessons will be taught using a modified direct instruction. Students will be given direct instruction with examples and ideas to guide their discoveries, but the majority of problem solving used in the examples will come from student input. If there are conflicting opinions on the next steps for the examples, students will discuss and decide how to proceed. Instructor assistance will be available and given frequently to aide students during assignment work time.

Write expressions in equivalent forms to solve problems.

For the purpose of scaffolding the learning progression and building on the students’ knowledge, this learning progression will start with Section 8.1 Multiplication Properties of Exponents in the textbook, which is an introduction to exponent properties. Students should have some prior knowledge of exponents and exponent properties as they are introduced in the 8th grade. The class will begin with an entrance activity to review the basic properties of exponents. These will be gone over in class, as it has been a while since students have dealt with exponents. Students will be given the generalized forms of one of the properties and then given an example, they will be asked for input as to how the property is interpreted and how it can be used to solve the problem8.EE.A.1. This will be repeated for each of the three properties, and then the students will be given a few more complex examples that use more than one property and the same student led solving of the problem will be used to solve them HSA.SSE.B.3.C. Students will be given a homework assignment that is a mix of problems where they will be asked to either evaluate or solve exponential equations. If students are asked to Evaluate, they will need to reduce the equation or expression to the simplest exponential form. When students are asked to solve, they will need to reduce all expressions to simplest exponential form and if there are numerical bases that have exponents, these will need to be reduced further. These definitions will be discussed when the assignment is given as it is important that the student produce the type of answer the problem requests MP6.

CCSS.MATH.PRACTICE.MP6: Attend to precision.

8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions.

HSA.SSE.B.3.C
Use the properties of exponents to transform expressions for exponential functions.

The second lesson in this learning progression will begin the next section in the textbook, Section 8.2 Zero and Negative Exponents. This lesson will start with an entrance activity where students will review problems using the properties from the previous lesson. The instruction will begin with students filing out an exponent chart from 25 down to 21, we will then discuss and look for a patterns, multiplying by two is a common pattern with increasing exponents, but when the exponents are decreasing it is divide by 2, we will then use this to find 20 and again with 2-1 and so on and look for a pattern in he numbers under the radical discovering the negative exponent rule. We will repeat this with a base of 3 and a base of 0 to determine that this rule works for all numbers except for 0, as well as discovering that any base to the 0 power is equal to 1. Students will be given several examples that will be worked through in class led mostly by students to help them see how the negative exponents can ‘jump the bar’ to become positive, and that we do not want to see negative exponents in the final answers HSA.SSE.B.3.C.. They will also have an example of how to create a chart for an exponential equation and be shown how to graph itMP4. Again students will be reminded of the difference between evaluating an expression with exponents and solving an expression or equation. This again is extremely important, as their answer needs to be in the correct formMP6.

Benchmark Assessment for the second lesson:

-Sketch the graphs of y = 3x and y = (1/3)x and use them to predict how the graphs of y = bx and y = (1/b)x are related.

CCSS.MATH.PRACTICE.MP4Model with mathematics.

CCSS.MATH.PRACTICE.MP6: Attend to precision.

HSA.SSE.B.3.C
Use the properties of exponents to transform expressions for exponential functions.

The third lesson will again begin with another entry task based on the previous lesson. This lesson will be based on Section 8.3 Division Properties of Exponents. Much like the first lesson in the progression, students will be given a generalized version of the quotient property of exponents and expressions they will work through together with the teacher as notesHSA.SSE.B.3.C. They will also be given a couple examples for their notes and finally be asked to solve a rather large multi-step exponential problem using many of the properties of exponents. Students will work through this together out loud with the teacher on the boardHSA.SSE.B.3.C correcting themselves and each other as they goMP6. They will then be given their assignment, reminded again of the difference between evaluating and solvingMP6 exponentials.

CCSS.MATH.PRACTICE.MP6: Attend to precision.

HSA.SSE.B.3.C
Use the properties of exponents to transform expressions for exponential functions.

The fourth lesson in the learning progression will be Section 8.5 Exponential Growth Functions(Section 8.4 will be skipped and returned to later). The lesson will begin as always with an entry task over the previous lesson. As notes for this section, students will be given examples of the exponential growth model and discuss each piece of the equation, what they mean and how to find themMP4. The class will solve several real-life examplesMP4 in the notes. These examples will include growth and decay functions and the difference in the equations and how to determine which HSF.IF.C.8.B will also be included in note material. Students will be then able to get some practice with the assignment where they will be asked to make tables of valuesMP4, determine function models from given information HSF.IF.C.8.B, and graph them if necessary.

CCSS.MATH.PRACTICE.MP4Model with mathematics.

CCSS.MATH.PRACTICE.MP6: Attend to precision.

HSF.IF.C.8.B
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and classify them as representing exponential growth or decay.

Benchmark Assessment for the second lesson:

-Given a graph of the length to diameter of a bluebird, determine which model represents the graph, estimate the length of an adult bluebird based on your life experience, use the exponential growth model to estimate the length of an adult bluebird, how do these compare? Is this a good model?

 Apologies that Benchmarks for lesson 1 and 3 will not copy into this post, but can be found in the original document here: Learning Progression 8.1-8.5

Modeling a Digital and Global Age Learning Environment

Modeling a Digital and Global age Learning Environment

Brittany Moore

Bolded words are linked to sites!

Technology is even more present in our world than ever, and a very large part of our students’ daily lives and in how they see and interpret the world around them, so it only makes sense that it is incorporated into their standard learning regimen.

Web 2.0 refers to the new stage in development in the World Wide Web that allows users to interact with webpages rather than just use them as a source of information. This is a wonderful opportunity for teachers and should be taken advantage of.

There are many new ways for teachers to connect and share ideas, which they can use to create new lessons and gain new teaching strategies or to find ideas to supplement their already tried and true methods. Teachers may also find this helpful to help students who are struggling and not grasping the concept the way the teacher is teaching it. Many teachers think of Pinterest pinterestwhen they think of turning to the internet to find creative lesson ideas, but would you have thought to look at Scholastic for interactive math lesson ideas? They even have some awesome tips for integrating technology into your classroom.

You have used technology find ways to enhance your lesson, now how can you present this information to your students in a way that uses the technology to make it fun and interesting? Rather than standing at the front of the room lecturing, one go to form of presentation is the Prezi, an interactive slideshow presentation that is customizable to your presentation. Now, a slideshow presentation may seem like a difficult or dull way to learn math, but this Prezi about the number system provides an amazing visual for students.

https://prezi.com/n-8ibkiw881r/number-systems/

Prezi can be accessed for free, but creating a free presentation allows Prezi to add it to their database for other users to search and view or use, which can also be a helpful way for teachers to find new ideas or ways to present topics. This could be an example to students about how to be responsible about what they post because once it’s online it stays online forever, and also to discussions about plagiarism the consequences of taking someone else’s work to pass it as your own.

Other sites that can be extremely helpful, Desmos Graphing Calculator is not only an extremely useful resource for students, it also has pre-programmed examples under the tab on the left side and allows sliders to be able to plug in variables into generalized formulas.

Screen-Shot-2012-12-08-at-8.01.45-PM

This could be used as a modeling exercise during instruction. A teacher could even project the graph or lines up onto the whiteboard and mark points or lines on the perfectly projected graph rather than having to draw the axis and try to guess while plotting points, this will make it easier for students to see exactly where points are. This is also a helpful resource for students when they are at home they can access the examples. Desmos is also available as an app on all devices and could be used as a support or accommodation for students who may have difficulties with fine motor function and are unable to graph on paper.

One way that utilizes technology and makes assessment fun for students is Kahoot!, a survey type quiz program that allows students to use their phones or computers to answer questions projected onto the board. Students receive points for their answers correctness and then the speed of their answer. This can be a fun formative assessment as well self assessment for teachers and students to gauge the students’ understanding of the topic and determine what materials need more focus or review.

Keep It Warm or Cool It Down HSF.BF.A.1.B

Keep It Warm or Cool It Down

Making connections between content areas is an important to our understanding of the concepts. This lesson would be ideal in an 8th grade algebra class where students also taking physical science, or in a high school algebra or functions class if students are taking physics. If students are not taking a physical science class and/or have not gotten to the topic of conductors this lesson can still be done with a limited understanding of the physics content. This project would take place during the a segment on exponential functions.

Standards for this project:

CCSS.MATH.CONTENT.HSF.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

CCSS.MATH.CONTENT.HSF.BF.A.1.B
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

CCSS.MATH.CONTENT.HSF.LE.B.5
Interpret the parameters in a linear or exponential function in terms of a context.

CCSS.MATH.PRACTICE.MP4 Model with mathematics.

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

 

General Equipment:

  • Vernier Temperature probes
  • Computers with Logger Pro

Vernier_EasyTemp_large

OR

  • Vernier Go Wireless Temperature Probes
  • Go Wireless Temp App for Ipad or other tablet
  • Aluminum soda can for each group
  • Boiling or VERY warm water

 

Specific Equipment:

  • Student chosen method of insulation/cooling for their can.

 

The Project

Students will be placed into an even number of small groups or pairs, half of the groups will be tasked with cooling the hot water in their cans to room temperature as quickly as possible and the other half will be tasked with keeping their water from cooling to room temperature as long as possible by only altering the environment outside of the can. Students will use their knowledge/minimal research to come up with plans in their groups and have the next day or two to prepare outside of class with their group gathering supplies, or building their cooling or insulating apparatuses.

Once the cans are prepared the instructor will pour very hot water into the cans, and students will put in the temperature probes. Have every group start recording their data and when the temperature cools to 90°C so all starting temperatures are the same. While students are doing this, allow a control, just a plain can to be going at the same time. Students will have a graph of their temperatures, and will need to find an equation to model their equation.

LoggerProMain

Students can write their equations on the board and use reason and their knowledge of exponential functions to determine the order that the groups cans cooled to room temperature. The functions can then be graphed together to model and compare the rate of cooling for each groups’ can.

Stainless Steel Temperature Probe

Vernier Logo

High School: Functions Linear, Quadratic, & Exponential Models*

This learning progression will take place in a 9th grade Algebra class at Ellensburg High School. These lessons will align with the mathematics content standards HSF.LE.A.1-3. The learning progression will also be aligned with these mathematical practices: MP4 Model with mathematics, MP6 Attend to precision, MP7 Look for and make use of structure.

 

Students have spent the year learning about and working with linear functions. They have used their skills and knowledge to find linear representations for data from charts and graphs, For this learning progression they will be using their skills to reason and problem solve to find equations that represent different types of data, pictorial or symbolic and numerical sequences as well as from graphs and charts. The students will use old and new skills to discover not all patterns can be represented by linear functions and will discover and form exponential and quadratic functions to represent the data where linear functions fail.

 

These lessons will be taught using cooperative and inquiry-based learning. Students will be given direct instruction with examples and ideas to guide their discoveries, but the majority of the learning will take place in small groups where students can guide one anther through their thought processes, self assess and self correct as well as assess and correct each other. Instructor assistance will be available and given frequently to aide and guide thought processes and steps to find equations.

 

Construct and compare linear, quadratic, and exponential models and solve problems.

Lesson One

CCSS.MATH.PRACTICE.MP6: Attend to precision.

HSF.LE.A.1.A-C
Distinguish between situations that can be modeled with linear functions and with exponential functions.

For the purpose of scaffolding the learning progression and building on the sudents knowledge, the learning progression will start with graphs and charts, content and problems similar to that which they have mastered. Before breaking into groups, the class will go over some vocabulary necessary for understanding one another’s’ thoughtsMP6 in this type of group discussion, such as: sequence, pattern, number in sequence n, function, equation, and the class will also discuss and come up with some rules for how they will speak to each other when if and when they disagree. For their first activity, students will take a list of data from a table and graph the points and find the equation of the line through the points as they have done before to determine the data is linear, as they have done before in previous lessons. Next students will be given pictorial sequences, where they will be asked to relate the number of the image in the sequence to the number of pieces or shapes that make up the sequence, students may find patterns to create equations, or to make a table of the numbers and graph to find the equation. The purpose of the bench mark assessment for the first lesson is to see that when given a non-linear data set, students can distinguish that a linear equation will not represent the data, while recognizing that there is a pattern to the dataHSF.LE.A.1.A-C.

Benchmark Assessment for the first lesson: Graph the data

x y
1 2
2 4
3 8
4 16
5 32
6 64

Can this data be fit with a liner function? Could it be represented by another type of function? Why or why not? Explain your reasoning.

 

Lesson Two

CCSS.MATH.PRACTICE.MP4Model with mathematics.

CCSS.MATH.PRACTICE.MP7Look for and make use of structure.

HSF.LE.A.2
Construct linear, quadratic and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

The second lesson in this learning progression will begin with a student led review of the pictorial sequences from the previous day, to address misconceptions and make sure all students are at similar levels of understanding. Instructor will introduce math tiles to aide students in finding patterns in the sequencesMP7. These tiles will be helpful to the student because this day’s sequences will not be linear and, while students know how to approximate these with linear functions, they will be asked to identify patterns and build functions that perfectly fit these seriesHSF.LE.A.2. These will turn out to be quadratic functions. After students work together using the tiles, they will be given a data table. This is more difficult to do with tiles so they may wish to graph the points and compare to the previous quadratic graphs and equations they have found, and work out the equation by comparison and using tilesMP4. Instructor help and hints will likely be a big aid for many groups, guiding questions and nudges in the right direction should be enough to help students get to correct answers. The Benchmark Assessment for the second lesson is to assess whether or not students can describe using precise mathematical language, describe a pattern they found in a quadratic sequence and how it differs from a pattern found in a linear sequence.

Benchmark Assessment for the second lesson: Using precise mathematical language, describe a pattern you noticed in a quadratic sequence and how it differs from the patterns of a linear sequence.

 

Lesson Three

CCSS.MATH.PRACTICE.MP7Look for and make use of structure.

CCSS.MATH.PRACTICE.MP6: Attend to precision.

HSF.LE.A.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

The third lesson will again begin with a student discussion of their knowledge of quadratic formulas, and go over the benchmark assessment from the previous day. Then, students will again work with their group to find equations to fit a set of data; this lesson’s equations will be exponential. Students will be given a data table to find an equation for. They will have the use of the math tiles again, to help them find the patternHSF.LE.A.2 MP7, but they may find it easier to see a pattern between numbers rather than shapes. Again, instructor help will be available to the groups, but by this lesson the students should be more comfortable trying new things and working together to solve a problem. The purpose of the benchmark assessment is for students to be able to compare the three different types of functionsHSF.LE.A.3, and describe their understanding using the vocabularyMP6.

Benchmark Assessment for the third lesson: On a sheet of graph paper, graph the functions in different colored pencils:

f(x)=2x

f(x)=x^2

f(x)=2^x

Discuss using precise mathematical language the behavior of the lines (higher, lower, or equal to) in relation to each other before x=1, at x=2, and after x=4.

Download the document here:

Learning Progression Moore

Volume in Sports

 

balls

Mens Basketball diameter 238.8mm

Soccer ball 110mm

Volleyball 105 mm

Baseball 37mm

Cricket 36mm

Tennis 33mm

Golf 12.35mm

Squash 20mm

Most sports throughout history have been played with some manipulated object, and most popular of all, is the ball. The object of this activity would be to compare the volumes of a variety of different sports balls. We can find the ratios between the different sized sports balls and find equations to represent the relationships between their volumes.

volume

For example, the volume of a soccer ball is 5575279.67mm3, and the volume of a squash ball is 33510.3mm3, and the ratio of squash balls to soccer balls is 116.375:1, or perhaps that a soccer ball could hold 116 squash balls (theoretically).

We could also find equations of how many golf balls and squash balls combined would fit into a basketball

(x*Vsquash)+(y*Vgolf)≤Vbasketball

  CCSS.MATH.CONTENT.8.G.C.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

F.LE Patterns to Quadratic Equations

Common Core State Standards

CCSS.MATH.CONTENT.HSF.LE.A.2

CCSS.MATH.CONTENT.HSA.CED.A.1

Task

Generalize the pattern by finding an explicit quadratic equation for the number of shapes that make up any given term, n, in the sequence. Let Tn represent the number of shapes that make up the nth term. Show your reasoning, then find Tn for the next term, then draw it to check your answer.

1.

n=1                  n=2                     n=3

Problem 1

 

2.

n=1     n=2              n=3                  n=4

Problem2

 

Commentary and solutions are available on the original document

here:  IM Task

 

 

Love Math with Desmos

Alex Heide, Brittany Moore, Nina Flanagan, Peyton Tamoyo

Desmos is a fully interactive mathematical tool that can be used for multiple concepts that allows for the students to be in complete control of their own curiosity.

What does Desmos do?

The ability to manipulate functions through graphs and tables is an important concept that allows for the students to visualize and to bring life the mathematical world that ranges from the lower levels of algebra to as high as advanced calculus and statistics. This application that is available on computers, tablets, and smartphones that can range from multiple Common Core State Standards that can be applied in multiple lessons within the classroom.

Built In Lessons

Desmos has 30+ built in lesson examples. In a specific lesson on function transformations, Desmos allows for the students to use sliders, a tool to switch between values interchangeably, to have power over what a variable will do to general function graphically. The slider tool is a unique characteristic of the program which enables students to input a general function such as a parabolic with scaler that you can hit play to animate the graph by changing the value of said scaler. The table option also give students another representation that gives specific coordinate points that can be useful for certain students’ learning needs. Students will be able to use visual representation to help guide them to understand how a scaler or scalers can alter the appearance of the function graphically.

Desmos helps to address student’s misconceptions of any math topics by allowing students to interactively change and manipulate the graph in real time. They can even press play on the sliders to allow the graph to continually for students to observe what happens for each variable.

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Classrooom Example

Desmos can aid in a lesson on transformations that addresses the Common Core State Standard CCSS.Math.Content.HSF.BF.B.3, where students build new functions from existing functions. This standard requires that students, “Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k.” Desmos will allow for students to use the sliders for each transformation case. This way the students will be in control of their own hypotheses on how the values of k will graphically manipulate the original function. For example, the teacher can ask the class what happens to the original function when we change the value of k? Students would be able to change the value of K using the slider and then come to the conclusion that k moves the function left and right. The teacher could also ask students to show a function that is translated up 5 and to the right 3. Students can do this in the calculator and physically show their work or save the graph to turn in later. Desmos has a pre-set example of a transformation and function that students can also use to adjust the sliders. This function can also be altered to any general function that is more applicable to the classroom’s knowledge.

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Desmos is a very interactive application that can be used in many facets of the classroom to enable and enhance student learning. This application is versatile and creates options for the teachers to project it from computers for a lesson or allow the students to work independently and/or collectively to generate their own conclusions. Desmos is very intuitive and user-friendly that it can be integrated into any classroom that will allow you to scaffold consecutive mathematical concepts and lessons.

Download the article Here: Desmos