Rebecca Zook - Math Tutoring Online

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Topic: how to solve math problems

An easy way to remember the difference between a line with zero slope and a line with no slope

Monday, October 8th, 2012

A lot of students get the concepts of “zero slope” and “no slope” confused when they’re first introduced.

Most students think something along the lines of, “They’re the same thing, right? Because zero equals nothing…..?????????? Wait… no, they’re totally different — BUT HOW DO I REMEMBER WHICH IS WHICH?”

Here is a super easy way to remember the difference:

Zero slope means that the line is horizontal. Just like the line that makes the top of a “Z” is horizontal.

No slope means that the line is vertical. Just like the line that makes the beginning of a “N” is vertical.

(If you’re interested in a mathematical explanation to go with the visual reminder, check out Elizabeth Stapel of PurpleMath’s lesson on slope. The part about zero slope and no slope is towards the bottom of the page.)

Many of my students have used this tip with great success — so spread the word! No one needs to be confused about this anymore!

Do you wish someone would just explain math in a way that really makes sense to **you**? Do you yearn for the confidence that comes from really GETTING it?

Just click here to get started with your special application for my one-on-one math tutoring programs. Once your application is received, we’ll set up a special phone call to get clear if my approach would be a good fit for your child.

I’m looking forward to hearing from you!

Related posts:
A visual way to solve elapsed-time problems
Gallon Man to the Rescue!
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Topic: how to solve math problems

How to convert from standard form (Ax+By=C) to slope-intercept form (y=mx+b)

Tuesday, May 8th, 2012

Here are two examples worked out of how to convert from standard form (Ax+By=C) to slope-intercept form (y=mx+b).

This is something that you get asked to do a lot as you start to get more comfortable going back and forth between different equations of a line.

And another example, because it’s nice to see more than one example when you’re learning something new:

If what you see here resonates with how you like to learn, and you’re looking to work with someone one-on-one to really master math, then I invite you to apply for my super special one-on-one math tutoring programs.

Just click here to get started with your special application for my one-on-one math tutoring programs. Once your application is received, we’ll set up a special phone call to get clear if my approach would be a good fit for your child.

I’m excited to connect!

Sending you love,

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Topic: how to solve math problems

How to navigate the space-time continuum (or, a visual way to solve elapsed time problems)

Wednesday, January 5th, 2011

When I was growing up, I learned to do elapsed time problems by subtracting the start time from the end time (and when necessary, borrowing while keeping in mind that there are 60 minutes in a hour).

This technique always seemed convoluted, so when a fifth grade tutoring student of mine was working on elapsed time problems, I tried this visual way of solving them, which seemed to be much more intuitive for my student.

Let’s say you have a problem like this: A train departs at 2:55 and arrives at 5:18. How long is the train ride?

First, draw a timeline:

Draw in the beginning and ending times, marking the hours as you go:

Draw loops to count the hours:


Add up the hours:

Draw loops to count the minutes left at either end:

Add the minutes together:

Combine the hours and the minutes, and you’re done!


That’s it!

Once you do a few of these, there’s lots of different ways to draw the loops. For example, you could start at 2:55 and loop to 3:55 (1 hour), 4:55 (2 hours), and then from 4:55 to 5:18 (4:55 to 5:00 is 5 minutes; 5:00 to 5:18 is 18 minutes; 5 + 18 = 23 minutes) to get 2 hours and 23 minutes.


And after drawing some of these out, you can use the same process to do elapsed time problems mentally, too. Just think about counting from one time marker to another and adding up the different loops.

*Visiting from this week’s Carnival of Homescooling? Welcome, I’m glad to see you here! Below are some more posts you may enjoy.

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Topic: how to solve math problems

Gallon man to the rescue!

Thursday, June 3rd, 2010

Do you need a way to remember unit conversion effortlessly and forever? Or just a way to calculate how many cups there are in a gallon?

Here’s how to figure it out. Draw a gallon man!

First, draw a really big capital G. (This is the gallon.)

Inside the G, draw four big Qs. (These are the quarts.)

Inside each Q, draw two Ps. (These are the pints.)

Inside each P, draw two cs. (These are the cups.)

For the final flourish, draw an arrow to one of the cs and write “8 ounces.” (There are eight ounces in every cup.)

When one of my students, a fifth grader, taught me about Gallon Man, I thought, I wish I had learned about this in fifth grade! My entire life, I’ve had to look up each of the conversions and never really internalized how they all fit together.

Since I’ve been introduced to Gallon Man, I’ve gleefully shared him with a fourth grade tutoring student (online), a friend who is a professional organic farmer (in person), innocent bystanders (at a restaurant), and most recently, my Mom (over the phone…”first, draw a really big G…”)!

They’ve all found Gallon Man helpful. Responses have included: “Can I take that drawing home with me?”, “Oh…I get it!”, and “I’m going to hold onto this.”

Gallon Man is totally visual and works for many learning styles. You can SEE how many quarts are INSIDE a gallon. Gallon Man is intuitive for all grade levels (unlike dimensional analysis, you don’t have to worry about the numerators or denominators). Gallon Man is practical. You can use it in your kitchen or in the grocery store. Gallon Man is easy to remember. And Gallon Man is fun to draw!

Gallon Man has recently gotten some airtime from other math bloggers, including Sam J Shah, who pointed out that it really helped him to see someone drawing Gallon Man. Here’s Sam’s post and video.

Yay for mnemonic devices!

*Are you looking for an online math tutor who uses multisensory methods? I’d love to help! Give me a call at 617-888-0160 to discuss your situation.

*Visiting from the Math Teachers at Play Carnival (Adventure Edition)? Welcome, I’m glad to see you here! Below are a few other posts you might enjoy!

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Topic: how to solve math problems

An Easy Way to Remember How Logarithmic Notation Works

Saturday, April 10th, 2010

Here’s a way my students and I developed to help remember what goes where in logarithmic form.

Many of my students have found this really helps them remember logarithmic notation!!


While this memory device is no substitute for understanding conceptually how logarithms work, it is very useful to be able to remember how to “rearrange the furniture” to change an exponential equation into a logarithmic equation.

And speaking of logarithms, I also highly recommend Kate Nowak‘s post on how to introduce logarithms without freaking students out.

Maria Droujkova of Natural Math also has a great post on how you can use family trees to demonstrate how logarithms work.

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Topic: how to solve math problems

GeekDad on Math Homework Mind Meld

Friday, February 19th, 2010

I’m super excited!! Curtis Silver has posted his response to my homework help tips, “doing the math homework mind meld with your geeklet,” on Wired’s GeekDad blog!! Thanks, Curtis, for your thoughtful response!

Curtis’s use of the term “mind meld” made me laugh, but also brings up an important point. As a tutor, part of what I’m trying to do is to make my students more like me—to make them more persistent, better problem solvers, and more active learners.

But in order to accomplish this, I frequently make myself more like my students. Do you like visual explanations? Let’s draw it. Do you like to see an example? Let me show you five examples. Do you have a question? I will answer it 200 times until it is crystal-clear. I believe in “more of what works.”

What intrigues me is that it is such a two-way street. I expand my students’ tool kits by making them more like me, but they also expand my tool kit as a teacher and problem-solver by forcing me to consider solutions that I never would have seen without them.

The paradox of “mind meld” is that in order to “mind meld” with someone, you need to understand how your minds are different in order to become more similar. The differences are actually what unleash the potential for change and learning.

Also, I’m glad that Silver highlighted continuous interaction, which is a huge part of my tutoring philosophy. But I want to clarify something important.

Sometimes, when a kid gets in the “math zone” and is confidently solving a problem without making any mistakes, I’ve found it’s totally appropriate to say nothing at all.

You still give the kid your absolutely undivided attention and watch their every move to make sure they stay on track. But when a kid is on a roll, interrupting them for the sake of being interactive – even just to praise them – can be counterproductive.

Sometimes being involved as much as possible means giving your kid undivided attention while staying quiet until it’s time to speak up.

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Topic: how to solve math problems

Tips for How to Help Your Kid with their Math Homework

Tuesday, February 16th, 2010

Back in October, GeekDad’s Curtis Silver postedabout a recent survey which found that parents would rather talk to their kids about drugs than about math and science.

Some parents don’t feel comfortable explaining math to their kids because they don’t understand it themselves. Other parents, even if they love math, find today’s teaching methods so different from how they learned math that they don’t know how to help.

Silver observed that even parents with advanced math knowledge might not know how to relate it to their own little geek. What’s a full-grown geek to do?

Wherever you fall on the spectrum of geekdom—whether you use math daily as part of a geek job, or haven’t done long division in decades—and no matter your kid is learning the times tables or studying trigonometry—here are some tips on how to help your kid with their math homework.

First, some thoughts on attitude:

Be an explorer, not an expert. Go into your math time in the spirit of a shared exploration instead of feeling like you need to be an expert. You can help your kid a lot, even if what they’re doing is initially unfamiliar to you. Don’t be afraid to say, “Let’s figure this out together,” or “I haven’t done it this way before. Can you tell me more about it?”

Stay positive and keep trying. Getting good at math means being willing to persevere in the face of a challenge. If you don’t get it right away, that’s OK. Kids learn a lot from watching someone model what to do when they’re faced with unfamiliar material.

Follow your kid’s lead. Just because your kid is the fruit of your loins doesn’t mean that their brain works anything like yours. So share any tips or tricks that work for you, but don’t take it personally if they don’t click with your kid.

Likewise, if your kid spontaneously comes up with their own learning strategy or memory trick, run with it. It will boost your kid’s confidence in their own thought processes.

Now for some nitty-gritty step-by-step suggestions:

Before diving in, ask your kid to tell you what’s going on. They get a chance to demonstrate what they do know, and you get a chance to review the material.

Also ask your kid to tell you what they don’t understand so they can reflect on their own learning and maybe pinpoint the missing links. (If they can’t articulate which part they don’t understand, that’s OK.)

Ask questions to walk them through the problem. Even if you understand the problem perfectly, don’t give a demonstration that puts your kid in the role of a passive observer. Instead, use really simple questions (that your kid has a 95% chance of answering correctly) to walk them through the steps of solving the problem.

Instead of telling your kid, “4 times 8 equals 32,” ask them, “What is 4 times 8?” Instead of telling them, “For the next step, we need to …”, ask them what happens next.

Asking questions keeps your kid from spacing out. It breaks down the process into smaller pieces. And the questions show your kid what they should do when they’re alone.

Plus, asking questions helps you find the disconnects. If you ask your kid, “What is 4 times 9?” and they say, “twenty-five,” you know you need to review multiplication facts.

When in doubt, write it out. Encourage your kid to write out all the steps in their work. The less they have to keep track of in their head, the more accurate they’ll be. And this lets you see their thought process.

Review an example from their textbook or handouts together.
If you don’t understand how to do a problem, and you have examples of problems being worked out step by step, go over a couple together.

Once you feel confident, practice by working on similar problems. To make sure you’re on the right track, try to pick problems where you can check the answer in the back of the book.

Backtrack. You can’t build a solid foundation on shaky ground. If there’s some prerequisite knowledge that doesn’t make sense to you, go back through your kid’s materials to find where the concept or procedure was first introduced and then review it together.

Or, if your kid understands it and you don’t, ask them to explain it to you, or take time to review it on your own.

Consult different materials. If the examples from textbook or handout still don’t make sense, do not despair. Seek another explanation from an alternative source. This lets you model resourcefulness.

For an awesome basic algebra text, try Algebra: Structure and Method. More visual or tactile learners might appreciate the Math U See curriculum. Girls learning or reviewing pre-algebra might enjoy Danica Mackellar’s excellent math books for girls, Math Doesn’t Suck and Kiss My Math.

Also, Khan Academy offers great instructional math videos in an organized index. YouTube has even more math videos, but it can take some digging to find the good ones. And you can check out some instructional math videos I made here.

In conclusion: You can help your kid, no matter how remote their math homework might seem to you. These steps can help your kid hone their very own math powers.

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Topic: how to solve math problems

When in doubt, talk it out

Monday, February 15th, 2010

Here’s a great new tidbit from my favorite magazine, The Week:

If you find yourself struggling to solve a complex math problem, try working through it out loud, says Scientific American. Psychologists in Spain found that college-level math students who detailed their thinking processes aloud were able to solve the problems faster and with greater accuracy than their silent counterparts.

In the study, quiet and nonquiet students were placed in separate rooms, given problems to solve, and monitored on videotape. The test results confirmed that students who talked aloud, or who drew pictures to map out the problems, scored higher and finished faster.

The researchers aren’t quite sure why this approach works, says psychologist Jose Luis Villegas Castellanos, only that representing a problem verbally or visually clearly offers “more possibilities of finding the right solution.”

This new finding makes me think of all the times in high school that I’d approach my math teacher to ask for help, only to suddenly realize exactly what I needed to do as soon as I started to explain why I was confused. I’d joke with my teachers about how they radiated understanding so I’d just “absorb” it once I was in their force field. But now I’m wondering if it was actually the process of getting ready to tell someone what I didn’t understand that activated my own inner knowledge.

This new finding also potentially explains why tutoring can be so powerful. In most math classes today, students passively receive information by listening to a teacher present the material to the class and then approach math problems in silent solitude at their desk. Talking things through out loud isn’t encouraged.

But in a tutoring situation, students are forced to talk things through out loud with their tutor. Maybe the process of learning to talk things out is as powerful as the process of “getting help” from someone who is more experienced.

I wish that more people were encouraged to talk things out and draw pictures to solve problems in standard math classes.