Teaching Algebra When Multiplication Is a Roadblock

 It’s a common and often frustrating scenario for math teachers: you're ready to dive into the exciting world of algebra, but a significant portion of your students struggle with the prerequisite skill of multiplication. Asking a student to solve $3x=12$ when they don't know that $3\times 4$ equals 12 turns an algebra problem into a fundamental arithmetic roadblock. The solution isn't to put algebra on hold, but to creatively bridge this gap by teaching the two skills simultaneously. Instead of seeing this as a problem, view it as an opportunity to build a deeper, more conceptual understanding of both subjects

The first step is to move beyond the traditional flashcard drills that may have failed these students in the past. We need to teach the meaning behind multiplication, not just the memorized facts. Use visual models to make the connection explicit. For example, when introducing multiplication, use an array model. Show that $3\times 4$ is a simple grid of three rows and four columns, with a total of 12 squares. This approach connects multiplication to the concepts of area and groups, making it a tangible idea. When you transition to algebra, an equation like $3x=12$ can be framed as "a rectangle has an area of 12, and one side is 3. What is the length of the other side?" This makes the variable x a concrete, understandable missing piece of a puzzle.

Allowing students to use the right tools can be a game-changer. For a student who is still learning multiplication facts, a calculator is not a crutch, but a tool that allows them to focus on the algebraic concept. Let them use it to solve $12÷3$so they can concentrate on the primary goal: understanding the inverse operation needed to isolate the variable. This builds confidence in their ability to perform the algebraic steps without being bogged down by an arithmetic deficit. You can introduce the balance analogy to explain the concept of inverse operations. Explain that an equation is like a scale, and to keep it balanced, whatever you do to one side, you must do to the other. To undo the multiplication by 3 on one side, you must divide by 3 on both sides.

Let's look at a few specific straggles to include in your daily lessons.  Begin by reinforcing the idea of fact families. If a student knows that $3\times 4=12$, they also know that $12÷3=4$. This simple mental connection helps them recognize that $3x=12$ is really just a division problem in disguise.

Next create embedded practice.  Instead of separate homework sheets, embed short, low-stakes arithmetic practice within the algebra problems. For instance, the first five problems of an algebra assignment could be single-step equations that require basic multiplication or division. This provides constant, contextualized practice without feeling like a remedial drill.

Reinforce "show your work" as a thought process.  Encourage students to verbalize or write out their thought process, not just the numbers. Ask them to explain why they chose to divide both sides by 3. This shifts the focus from the right answer to the right method, helping you identify and correct misconceptions about both algebra and arithmetic.

By integrating these strategies, you empower students to build foundational arithmetic skills within the context of more advanced topics, ensuring they not only learn algebra, but also gain the confidence and skills they need to succeed in math.  Let me know what you think, I'd love to hear.  Have a great day.

What Essential Concepts Are Needed For Success In Algebra?

 

Algebra is often perceived as a mysterious collection of variables and equations but it can open up a powerful world of problem solving. However, what concepts do students need in their foundational knowledge to navigate and master the topic.  Time to look at the key building blocks students need to succeed in algebra. 

First, students need to understand the power of variables. Variables, represented by letters like x and y, are the cornerstone of algebra. They act as placeholders for unknown values,allowing us to express general relationships and solve for unknowns. Understanding how to manipulate variables – performing operations like addition, subtraction, multiplication, and division – is crucial for success.

Next they need to see that equations are balanced. Equations are statements that declare two expressions equal. They are like scales – for an equation to hold true, both sides must be balanced. Understanding how to solve equations – finding the value of the variable that makes the equation true – is a core skill in algebra. This involves using addition, subtraction, multiplication, and division to isolate the variable on one side of the equation.

In addition, they should understand that expressions are the building blocks of equations. Expressions are combinations of numbers, variables, and mathematical operations. They can be simple, like 2x + 3, or more complex, involving brackets and various operations. Simplifying expressions – combining like terms and using the order of operations (PEMDAS) – is essential for manipulating expressions and solving equations.

Furthermore, they should know how the order of operations actually works rather than being able to recite the rules. PEMDAS, an acronym for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), dictates the order in which we perform operations within an expression. Following PEMDAS ensures we obtain the correct result when simplifying expressions and solving equations.

It is important that students know how to factor or expand equations. Factoring involves breaking down a polynomial (an expression with multiple terms) into its component parts. Expanding is the opposite – combining factored terms back into a single expression. These skills are vital for solving various types of equations and simplifying complex expressions.

Then they should understand how functions work. Functions are relationships between variables. They describe how one variable (the input) affects another variable (the output). Understanding basic function notation (f(x) for the output of function f evaluated at input x) and graphing functions is essential for many areas of algebra and beyond.

These core concepts form the foundation for success in algebra. By mastering these building blocks, students can progress to solving linear equations, tackling inequalities, understanding systems of equations, and venturing into more advanced topics like quadratic equations and exponential functions.

The journey through algebra may seem daunting at first, but with a solid grasp of these fundamental concepts,perseverance, and a willingness to practice, students can unlock the secrets of this mathematical realm and unlock a world of problem-solving abilities.  Let me know what you think, I'd love to hear. Have a great day.

Using Newspapers in High School Math

  I love the idea of using newspapers in high school math but where I live, we have access to a weekly one which arrives Wednesday afternoon if the weather is good or Thursday otherwise.

The good news is that many newspapers can be found online so access is quite reasonable if you are not near a daily one. Some newspapers do offer complementary copies to schools if wanted.

Newspapers offer the perfect way to connect classroom learning with the real world.  So what are ways you can use newspapers in your classroom?

1.  Use the weather maps to find mean, median, and mode.  Most newspapers have wonderful weather maps which may cover the area or the nation.

2. Take the temperatures and have students convert them from Fahrenheit to Celsius or have students convert miles per gallon to kilometers per gallon.

3.  Look at mpg for various cars and have students compare models to determine which model has better gas mileage in percents.

4. Check out sales flyers to compare prices to determine better prices and again use percentages.

5. Use inequalities to find the best buys for used vehicles < $3000 or >$35,000.  Apply inequalities to mileage for cars or perhaps use the same type of criteria for appliances, or even sports.

6. Have students write word problems to go with an assortment of graphs and tables.

7. Look at clothing, food, or other ads to write functions based on buying combinations such as jeans and shirts.  Or you could have them write functions for inequalities based on not spending more than a certain amount for your new jeans and shirts at the beginning of school.

8. Check out the help wanted ads so students can take a yearly salary and figure out the monthly or weekly gross pay.  If they find a weekly or monthly amount, students can calculate yearly salary.

9.  Create posters with real world examples of math to hand around the classroom.

10.  Go house hunting.  Find a house, then calculate the mortgage payment for 15 and 30 years using current rates.

11. Rank the colleges using the Academic Index which uses algebra to rank colleges. Change the parameters to see how that changes college rankings.

12. Check out this article which has lots of different ways to use the New York Times in Algebra.  Ideas 10 and 11 are from the article but they have several more great ways.  In addition, they include the appropriate standards and mathematical practices these activities meet.

If you want to include this article from 1922 which looks at the use of Algebra in Tariffs.  Image, even that long ago, they asked the same question regarding its use. 

Enjoy, have a good day and let me know what you think.

What Skills Are Needed For Calculus?

  While researching yesterday's topic, I discovered a short article or two on the skills needed for Calculus.  It turns out, students need a good foundation in both Algebra and Trig!

The specific Algebraic skills needed include factoring, completing the square, know the basic functions and transformations, simplifying expressions, manipulating formulas, and knowing how to use a calculator for graphing.

For Trigonometry, students need to know the three basic trig ratios and their graphs, the values for the basic trig functions and understand what constitutes an exact answer.


If they do not have solid foundation in the above topics, it will make it much harder to do calculus.  Since, I know these particular skills are the most important ones, I can focus on making sure my students learn these in Algebra I, Geometry, and Algebra II.  In addition, there are certain algebraic mistakes students repeatedly make in the calculus class.

So what are the mistakes students regularly makes?  Well -

1.  Students try to apply an additive property to any function with an addition sign, not just the linear functions.

2. Students often cancel parts of an expression rather than the whole expression.  I've seen students turn this - x^2 + 2/ x^2 -3 into  2/-3 because they've crossed out the x^2.  They have trouble understanding they cannot do it.

3. Students often confuse negative exponents with fractional exponents.  So sometimes they see 1/x^2 as x^1/2. 

4. Students often have trouble with exponential rules and sometimes apply them incorrectly.  For instance, you might see a student do this: 4(x - 1)^2 = (4x - 4)^2 which is not correct.

5.  Incorrectly simplifying using a value that is not equivalent to one or they try to simplify an expression rather than an equation.

6.  The order of operations and distributive properties when not followed correctly will create a problem.

These are really the six algebraic mistakes most students make in Calculus because their algebraic foundation is not as strong as it might be or the students do not quite understand the process.

If we know what the areas of Algebra and Geometry need to know well for Calculus and we know the most common mistakes they make, we can work on strengthening those areas so our students will be better prepared to do well in college. 

Solve for X

 The other night I stumbled across a nice little app called Solve for X.  This app is different from most of the apps I've run across in that it requires the student to tell the app the next step.

The app has six levels.  The first level has students solving one step equations.  You have to complete a certain number of problems correctly before you can unlock the next level.

The second level has students solving 2 step equations using addition or subtraction first, followed by the multiplication or division. It isn't until you get to the third level that fractions are involved. The interesting thing about this app is if you put in the wrong numbers, add 3 instead of subtracting 3, it will do exactly what you said and you can see what the results.

This app will allow my students to practice solving equations and I can use it to help reinforce and differentiate instruction.

Interactive notebooks

I"ve been reading up on ways of creating and using interactive notebooks in Math.  All of the ways I"ve found use a composition book and foldables to create the notebook.  I know I am trying to use more technology in the classroom so I can avoid using more paper and have more things my students have to keep track of in class.  So I am wondering if I could use a book creator app or even ibooks author so my students can create interactive math notebooks.  I need to research this a bit more as I want students to be able to bring in their drawings and work and not just rely on what they can import from the internet.  I am also wondering if they can design something that flips or moves for the notebook. 
On a different note, I had my seniors start using mind map for sin and cos graphing.  The areas are altitude, shift both horizontal and vertical, phase, and frequency.  So far, it is going well.  The one thing I learned is that it takes time for them to use the mind map app right now and we are only going to be able to add one or two strands a day.
I am hoping to have my algebra 2 students use it beginning Thursday for finding linear equations.  I have to make it Thursday because some students took the initiative to download iOS7 and one or two got messed up so I have to spend part of today getting the iPads up to date, imaged, and running.

Mindmapping

I was doing a search for using mindmapping in Math to get more ideas for using mindmapping in my math classes and ran across this site called mathsrevision from Scotland.  It has nice mindmaps that can be used by the teacher in the classroom using a smartboard and each thought or point will appear one at a time as you click the screen. I am still experimenting to see how well it works on the iPad. 
Once I redid the way I entered my search parameters, I finally got more of what I wanted.  I found you can use it for problem solving and the website gave lots of nice mindmaps. 
I am going to try it with my Algebra 2 later today to see how it works for finding equations of linear equations from graphs, with the College Prep math class for trig and maybe with the geometry class for the basics. 
I downloaded a free mindmapping app and I'll report back tomorrow on how it goes.

algebra game

I reported on an algebra application over the summer.  It has a blue background with a monkey on it and is written by Monkey in the Middle apps people. I had my first period class play the game and they enjoyed it. When you click on the first topic - words into equations, it gives you three choices, a game, flashcards or a timed challenge. The game has you helping a monkey go up the platforms and along the way you get 5 multiple choice questions.  If you get the questions correct, you keep going but if you miss the question, the monkey has to start again.  I had one student who did not want to quit when the bell rang.
I am finding games provide a venue where I can have students practice and reinforce skills which they need to work on and not having to take extra time to reteach the skills.  I like doing the games during the last 5 to 10 min of class.  One student even said "Where did the time go?" today when I asked them to clean up.  He is one who normally complains about time going too slowly.

Algebra Equation Builder app

Stumbled across this nice app.  It has 4 levels which increase in difficulty and requires people to take four terms and arrange them with two terms on each side of the equal sign so the equation itself is true.  Level one only uses numbers and when done, you move to the next level which includes a variable and tells you the answer.  Thus as students move through the levels, the difficulty increases until you have to come up with some rather complex problems.  I found some of the problems to be rather challenging and I think this game would give students a chance to practice without it being too easy.
Change of subject.  I got trained in Apple configurater so that next week I can start getting my iPads and the elementary school mini's set up and running.  So my prep periods will be spent getting the iPads up and running so I can actually begin using them in the classroom.

Math DJ into to Algebra app

I down loaded the free version.  Again the free version has only one chapter included with 30 problems to practice.  Each problem is for the most part a single step equation the student has to solve.  These problems are solved the same way as yesterday's app.  The student selects the correct number of bubbles with the letter so as to get the correct answer for the variable.  The problems start out easy but by problem 17 or so, they become much harder.  One example is 5G - 36 = -G which takes a bit more thought to solve.  If a student comes up with the wrong answer, it tells you that your answer is incorrect and to retry so a student must rework the problem till he gets it correct.  This app is one I could easily use in my lower  level algebra I class to help scaffold students who struggle with solving simple one step equations.  I much prefer this over the pre-algebra app for practice.

Algebra Tutorial app

I found this free app in the itunes store and it sounded quite good.  I opened it up and noted that it has lots of topics such as fractions, linear equations, inequalities, systems of equations, various equations for finding a line, circles, factoring, and quadratic equations.  I looked at a few topics such as systems of equations.  Each topic and subtopic shows examples of working problems with some written steps but the written step takes the place of actually showing the step visually so if you need to see the actual step, you are out of luck.  The app does provide the opportunity to look at more than one example. 
The other problem is that in some topics they only show one way of doing the work.  For instance, in systems of 2 equations, the app only provides examples for the elimination method.  I know that I will often tell students to use substitution if they can because it is a bit easier.  This app does not show substitution.
I am not sure this is an app I would use in class because the examples are not done as well as examples in a book.  I think this app is more for adults who are reviewing their algebra, rather than being used in class.  I have other apps, I can use to take examples out of the book, write questions for them to answer and have them submit the answers electronically. 

Barron's algebra app

This is the second app I found by Barron's but it is for algebra.  The first level is just reviewing the use of positive and negative numbers.  It has three. Quiz levels and you must get 80% in order to unlock the video game and move up to the next level.  Each level has 20 questions and again the program does not tell you which problems you missed.  This maybe due to not having a large base of questions to choose from.  

I think not telling students which problem they missed is not helpful but it does make them check all their work especially if they do not like figuring out where they made the mistake.  I see some positive uses for this app in my classroom.

Hup Algebra app

I found this line of apps by HUP.  The one I am looking at is HUP algebra.  It is a nice app which could be used to help flip the class and have students use the app at home.  It has several sections starting with "What is algebra?".  Then it has sections on solving single step equations using addition, subtraction, multiplication, division.  Each has a pictorial example using tiles, a written example and you can try problems.  There is a section on solving multi-step equations and on solving equations with the distributive property.  All sections have practice problems.  I like this app as I can use it to flip the class or I can use it to scaffold some of my students who need additional help or practice. It is free at iTunes.  I just got a Nook and I am starting to find neat apps there so I will review some of those.

More on centers

I have been giving some serious thought to using iPads with centers.  I have a book with printed instructions for using centers in Algebra I and ll, geometry and pre-algebra.  I can scan the various activities into PDF form.  If I have an app on the iPad that allows the students to read and write the answers on the file, they can then send it off to me so that I can grade it.  By using centers, I can increase their group activities and hopefully decrease their dependence on me. I look forward to trying this.  If I store the pdf files in the school wiki, I can use a QR code to send them to the proper page, provide instructions and any additional information.  I am trying to change the main focus from teacher oriented to student oriented.

Hands on equations lite

When I first downloaded this app and opened it I was a bit disappointed with it because it only had a video for each of the three levels.  All other functions were locked and you have to pay for an upgrade if you want access to all the functions.  After some thought, I decided it is not a bad thing.  I watched the the first video and liked the way it used a pawn and a balance scale to explain solving for the variable.  This gives students a new way to view the process and it is a way to differentiate my instruction.  In addition, I can have students rematch the video as many times as they need.  Hmmm I could use this app as part of a center.
I know they use centers in elementary school but not as much in high school.  I just realized I could use many of these apps as part of a centers set up.  I have a classroom set of iPads that I could set up into centers using the app.  I would have to arrange for instructions that would have students do more than just watch the video.  They might have to answer questions, do some of the problems out of the video.
If you have used mobile devices as part of centers, I would love to hear from you.

Algeboats lite

Found a nice little game called Algeboats lite for students to practice evaluating one and two step equations.  There are boats with equations such as x-3.  On the dock are four loads each with numbers you could substitute in for x and there are four flags at the top with the answer to the problem based on which value of x you choose.  Once the value for the variable is substituted and the correct answer is put on the flag pole, the boat moves away. There are three levels, rowboat, speedboat and yacht and each level has three sublevels which are unlocked as the first sublevel is completed.  The student can get a fair bit of practice.  The only thing that might frustrate students is that the loads and the flags do not always move as directed and it can sometimes take a while before moving.
I mention games because my students love playing games and this is something that could be done when there is time left in the class after they have completed the assignments or done after the weekly quiz on Friday or perhaps to obtain an exit pass.
Games provide possibilities for students, especially the lower performing ones, to practice in a fun and safe environment.

Factoring

As I was working on deciding exactly what my students should know in each class I teach, I realized that factoring is extremely important in more than algebraic fractions.  I discovered a nice game on prime numbers and prime factorization that also shows factoring in a way students can see how prime numbers are used as factors.  It is Prime Smash by Panasonic  and Risupia.  This game has three levels where balls bounce up one at at time and you decide if they are prime numbers or composite.  If they are prime you tap and if they are composite you swipe the number to split it up, tap a prime or swipe a composite and you break it down till it has been totally factored. Sometimes the program splits a number into a prime and a composite or two composite numbers.  You have to think quickly when you play this game.
It has a nice fun facts sections that defines prime numbers, defines twin primes, shows emirps and Palindrmic primes, and talks a bit about how to distinguish prime numbers and how they are used for incription.  This is a free app and a really nice one.
I am having fun on the easy level learning to play the game.  I've tried the medium level but its a bit fast for me quite yet.  This game is actually lots of fun once you get the hang of using it and I can see having students play this game on a regular bases.

Middle school math 7

I downloaded the grade 7 free app by the same people who made yesterday's app.  The first four topics are free and include adding and multiplying negative numbers, absolute value and order of operations.  Each topic has three levels and it keeps track of your score and your time.  Again the student can send results to the teacher. Both apps offer the student a chance to reset s ores and do the whole level again. In this game, the student helps the monkey move down a ladder, stopping every so often to pose a question.  I think this app is much more appropriate for middle school, pre-algebra, and lower performing algebra students.  I plan to have my algebra one students play these games over the first week or two of school to help them review after a summer off from school.  I hope my students like the app.