Tips and Techniques: Transposing

Transposing is an important technique for students to master as they progress from Algebra I to Algebra II and beyond. It’s the process of moving quantities across the equal sign in an equation to make the problem solving process more efficient. Here’s how it works.

 Solving an equation means isolating the variable on one side of the equation, either the right side or the left. Generally it is better to isolate the variable on the left side of the equal sign because we read from left to right.

Beginning algebra students are taught the Law of Equations:  whatever you do on one side of an equation, you must do on the other side as well. 

Using the Law of Equations, to solve 2x – 3 = 13, we can add 3 to both sides, giving us

2x – 3 + 3 = 13 + 3 ===> 2x = 16

Then, we divide both sides by 2 to find that x = 8.

To solve 2x – 3 = 13 by transposing, we do the following:

Instead of adding 3 to both sides, we can transpose the 3 that’s already there by

moving it from the left side, over the equal sign, to the right side, and changing its sign from “–” to “+.”

Now we have  2x = 13 + 3, which becomes 2x = 16, the same answer we got using the Law of Equations. Again, we find that x = 8.

The beauty of this process is that it is visually easier to see how the parts of the equation change as we solve it. Instead of writing the transposed quantity twice (once on the left and once on the right), we literally pick up the quantity and move it to the other side. The price we have to pay for doing this is changing the sign of the quantity being moved. 

Let’s solve 3y + 2x – 3 = 7 for y.

Since we want to isolate y, we can transpose 2x and – 3.

This gives us y = –2x + 7 + 3. Simplifying, we get y = –2x + 10.

If we used the Law of Equations, we would have to write out

3y + 2x – 3 = 7 ===> 3y + 2x – 2x – 3 + 3 = 7 – 2x + 3

While simplifying the above does give us y = –2x + 10, it comes with a lot more work.

Note that most classroom teachers will not allow transposing until students learn to use the Law of Equations.

Try these:

1. Solve for x: 4x + 9 = 49
2. Solve for y: 2y + 7 – 3 = 24
3. Solve for y: 3x + 9y – 7 = 49
4. Solve for a: 3b – 4a – 6 + 4 = 49
5. Solve for a: 2b + 4a + 5 – 2 = 42 – 2a

Jump Start Your Summer with Mathnasium and Math Games!

Yay, summer! As you and your kids enjoy a well-deserved break from the pressures of school, we’d like to remind you that summertime also presents many opportunities for kids to keep their math skills fresh and play around with concepts they may encounter next school year. In particular, integrating some time for math games into play dates or family nights encourages kids to flex their math muscles as they spend quality time with their favorite people! Here’s a list of our favorite math games, featuring a few classics as well as some fresh alternatives:

Monopoly

A fun, family-friendly board game in which players move around the board buying and trading properties, building up properties by buying houses and hotels, and charging the other game players rent. Play the game with “dollar” bills only—the debit card version is not as educational. 

Math War

Math War is a fun variation on the classic card game that challenges players to perform a calculation (in this case, multiplication). The player with the highest product wins the hand! 

Set

Set is a fun, portable (deck of cards) game that tests visual perception and pattern recognition by challenging players to choose three cards that all follow the same pattern or a different one.

Battleship

Battleship is an engaging game for older kids that tests memory and spatial reasoning skills. Based on a numbered and lettered grid, guess where your opponent’s game pieces are positioned. First one to “sink” all of their opponent’s battleships wins.

Are You Ready? Game Shows® (Available for K-5th Grade)

Is your child ready for the next grade? These games let kids become game-show contestants, winning pretend cash for each math or language question answered correctly. After playing, enter the game results online and get an instant online assessment that pinpoints areas where your child may need additional help. An added perk: Lakeshore Learning has a special offer just for Mathnasium: get $10 off any Are You Ready? Game Show! Enter code 9347 at checkout or click here for a store coupon. Expires 7/31/14.

PS: In search of more math-tastic summer activities? Our summer program is in full swing and everyone at Mathnasium is having a blast helping students fortify their math foundations and prepare for what lies ahead! Don’t miss out on the fun and learning—call or email us by June 30th to get a limited-time summer registration discount. For more info about Mathnasium, please call or visit our center's website. We look forward to seeing you soon!

April/May 2014 Math Muscle Challenge!

Can you solve this month's Math Muscle Challenge?

Grades 1 – 5: A recycling center pays $1.90 for every pound of aluminum recycled and 90 cents for every pound of plastic recycled. Anthony collected 10 1/2 pounds of plastic bottles and 13 pounds of aluminum cans. If he turns the bottles and cans in to the recycling center, how much money will Anthony get?

Grades 6 and up: The volume of a cube is 8 in³. Amy created a new cube by quadrupling the length of each side of the original cube. How many times larger is the volume of the new cube than the volume of the original cube?

Answers to Last Month’s Math Muscle Challenge:

Grades 1 – 5: 3 ½ miles.

Grades 6 and up: 60 regular subs

Tips and Techniques: Finding 100% of a Number

Given a percentage of a number, how do you find 100%? We’ll show you different ways to find 100%.  

First, remember that 100% means “all of it… the whole thing.”

Example 1: 25% of what number is 5?

This question asks, “If 25% of a number is 5, then what is 100% of the number (the whole thing)?”

The strategy here is to see how many times the “percent number” (in this case, 25) goes into 100, and then count by that number until we reach 100—the whole thing.

Here, we’re told that 25% of a number is 5. So, to find 100% of the number, we count by 25s up to 100: 25, 50, 75, 100.

25% is 5, so

            50% is 10,

                        75% is 15, and

                                    100% is 20.

Example 2: A man spent $10.00, which was 20% of his money. How much money did he have to start?

If 20% of his money is $10.00, then 40% is $20.00. (20% + 20% = 40%)

Similarly, 60% is $30.00 (40% + 20% = 60%), 80% is $40.00, so 100% is $50.00—the amount of money the man had to start.

Here’s an even faster way to approach this problem. There are 5 20%s in 100%, so this problem calls for five sets of $10.00, which equals $50.00.

Example 3: 10 kids—5% of the kids enrolled at a particular school—are out sick. How many kids are enrolled at the school?

Here, 5% is 10 kids. Since there are 20 5s in 100, we multiply 20 by 10 and get 200, the total number of students at the school.

Try these:

1. 10% of what number is 8?
2. 20% of what number is 4?
3. 20% of what number is 6?
4. 50% of what number is 28?
5. 5% of what number is 11?
6. 1% of what number is 2?

Seize the Summer: Perfect Time to Get Ahead!

As we close the book on the 2013 – 2014 school year, at Mathnasium, we’re excited about the possibilities for our students once summer rolls around! Here are a few reasons to seize every opportunity this summer – the perfect time to get ahead.

•  Everything kids do during the summer helps them get ahead. According to a recent study, at summer’s end, students perform one month behind their progress in the spring. Because new school material is at a standstill, everything kids do during the summer to build their math skills helps them get ahead!

•  Summer is also the perfect time to revisit and fortify math foundations. Math concepts build over time, and often, we find that students fall behind because they didn’t fully understand topics taught in earlier grades. While filling in these gaps is a primary objective of the Mathnasium program 365 days a year, without school stress or homework pressure to worry about, kids have even more time to step back and master these skills over the summer.

•  We love helping our students prepare for next year’s challenges. Another summertime perk: more time to get a head start on material they’ll encounter next school year. And the more prepared students feel, the more excited they’ll be about challenges that lie ahead. Third party studies show significant increase in grade level skills after three months at Mathnasium!

Mathnasium’s summer program offers a customized program for your child, flexible scheduling, and many opportunities to make this your child’s most productive summer yet! Call or email us by May 31^st to get an early summer registration discount. For more information, check out our website.

P.S. May is Teacher Appreciation Month! From all of us at Mathnasium, we’d like to say a big thank you to all teachers for giving young learners the gift of knowledge and for allowing us to support their math curriculum.

“If you are planning for a year, sow rice; if you are planning for a decade, plant trees; if you are planning for a lifetime, educate people.” – Chinese proverb

“Education is the most powerful weapon which you can use to change the world.” – Nelson Mandela

Math Test-Taking Strategies to Boost Confidence

Math anxiety can easily undermine the hard work students have invested throughout the year, lowering their scores on standardized tests and jeopardizing their chances to take advanced classes or enroll in the schools of their choice.

But reducing math anxiety, a well-documented phenomenon, can prevent this. Students can reduce anxiety by reviewing the mistakes they make on homework assignments and practice exams. After grouping these errors into three categories, they can easily be remembered as the “three C’s:”

• Concept, understanding the methods needed to resolve specific problem types
• Comprehension, determining exactly what individual problems are asking students to do
• Calculation, solving the problem correctly without making errors or overlooking any critical details

When faced with challenging test questions, students can become gripped with math anxiety.  However, they can address the root cause by reviewing past assignments and practice tests.  Often, the cause lies in difficulties with one or more of the three C’s.  After identifying where exactly a student’s weaknesses lie, adults can help address them.  This allows students to move on from anxiety and perform at the level of their true potential.

As a first step, parents, teachers or private instructors can sit down with students and look at the types of errors made in homework assignments, regular tests and practice exams.  It’s a process that can be helpful anytime during the academic year, especially when students become anxious about math, and is particularly important when it’s time to start preparing for standardized tests.

To address concept challenges, students can be asked to break questions down into a series of meaningful parts, resolve these parts individually, and then put them back together. Children struggling with both Concept and Comprehension can be asked to read troublesome questions aloud.  This helps to strengthen the mechanics of understanding by invoking different parts of the brain. Another helpful approach is to ask students to reframe questions in their own words.

To help students overcome Calculation difficulties beyond rote practice, which is more appropriate in this situation than the others, it’s crucial to make sure a child’s written work is neat and clear.  Neat penmanship makes a huge difference in kids’ ability to think clearly and follow their own good reasoning.

As the day of the test approaches, children should also take some common-sense steps that can be applied to all types of tests, Larry Martinek, chief instructional officer at Mathnasium notes.  First, avoid last-minute cramming by learning to pace yourself and structuring a daily study plan.  Make sure to eat a healthy, high-protein breakfast the morning of the exam.  Then, in the testing room, STOP and close your eyes.  Take a moment to inhale deeply.  When you exhale, open your eyes and envision the test with an “I can do” mindset.

Studies have shown math anxiety to impact up to half of all students in various ways.  However, with the proper approach, it can be effectively addressed and need not be a permanent hindrance to performance. 

Master Fractions and Set the Stage for Success, As Featured in The Wall Street Journal

Algebra Readiness begins in elementary school. Three essential topics in elementary school mathematics are 1) numerical fluency, which includes the effortless recall of addition facts and the times tables stemming from mental math strategies for computation, 2) foundational understanding of the nature fractions, and 3) the basics of problem solving. Mathnasium was recently featured in The Wall Street Journal and highlights our unique and effective approach to teaching fractions. Read the full article here.

As students move into middle school, work with fractions expands to embrace rational numbers (common fractions, decimal fractions, percents, ratio and proportion, and negative numbers). At the same time, student’s problem solving horizons broaden to include multi–step word problems, both routine and non–routine.

Successful mastery of these elements sets the stage for success in Algebra, the rest of high school math, and college. If any of these areas are under developed, the road ahead is fraught with danger.

Visit a Mathnasium Learning Center and see where your child stands.

- Larry Martinek