Introducing Make It Real Learning workbooks

I have recently had the pleasure to add Make It Real Learning workbooks to my site. These books contain real-life math activities with real-life data, companies, and situations. They are written by Frank Wilson.

Arithmetic I



Fractions, Percents, and Decimals I



Linear Functions I


Calculus I



Periodic and Piecewise Functions I

Some examples of the topics included in these activities are: cell phone plans, autism, population growth, cooking, borrowing money, credit cards, life spans, population growth, and music downloads. But there are many more, more than I can list here.

As students work through the problems, they can use the math skills and concepts they have learned in their math curriculum (such as the concept of average or graphing), and apply those to a situation from real life.

Each activity-lesson in the book contains several questions about the situation, starting with basics and going into more in-depth evaluations, and should be adequate for one-two complete class periods.

Why does that benefit you, the teacher? It will motivate the students far more than dull, boring word problems from the textbook. It will show students how mathematics is truly USEFUL. Also:

• The problems are written by an experienced math teacher (Frank Wilson)


• The problems are matched to the learning objectives of the National Council of Teachers of Mathematics (NCTM). This means that the concepts and skills required to complete the problems ARE found in typical middle and high school mathematics curriculum. You can simply replace some of the problems in your textbook with these real-life scenarios.
  
  
• These activities are excellent to be used in a small-group setting.
  
  
• Typically, the activities contain challenging parts and therefore allow students to practice real problem solving - not just apply knowledge from textbook examples to other almost identical problems.
  
  
• Gifted students can enjoy the challenge of solving all the questions on their own.

All books include complete solutions to all activities and problems. Please note that Make It Real Learning workbooks do not contain the instruction or explanations of the concepts.

List of available books:

Arithmetic I - for grades 3-6
Fractions, Percents, and Decimals I - for grades 4-8
Fractions, Percents, and Decimals II - for grades 6-11
Sets, Probability, and Statistics I - for grades 6-10

Linear Functions I - for algebra 1 and algebra 2
Linear Functions II - for algebra 1 and algebra 2
Quadratic Functions I - for algebra 1 and algebra 2

Exponential and Logarithmic Functions I
- for algebra 2/precalculus
Periodic and Piecewise Functions I
- for algebra 2/precalculus
Polynomial, Power, Logistic, and Rational Functions I
- for algebra 2/precalculus
Calculus I - for grade 12

Review of Kiss My Math and Math Doesn't Suck by Danica McKellar

Recently I had the delight of receiving a review copy of Math Doesn't Suck and Kiss My Math by Danica McKellar. Danica is well-known for her acting career, and her books are best-sellers.

First some basic info. Math Doesn't Suck covers middle school math topics: factors and multiples, fractions, decimals, percents, word problems, and a little algebra.

Kiss My Math covers pre-algebra: integers & negative numbers, variables and working with expressions, solving (linear) equations, word problems, intro to inequalities, exponents, and an intro to functions and graphing lines.

There's a website for each book: KissMyMath.com and MathDoesntSuck.com.

I've split this review into two parts: the pros and the cons. Let's go with the pros first.

Pros:

Danica's writing is excellent. It's conversational and catchy, very friendly. Reminds me of Dr. Math books.

I absolutely adore some of her illustrations that she uses for math concepts.

For example, she tells us to think about positive and negative numbers as breath mints - mint-egers. The positive ones have a good taste, and the negative ones are yucky, getting yuckier as they get more negative. And then she combines the integers as if they were mint-egers in her mouth! It's a fresh, fabulous comparison. You see, that'll stick to your mind for sure... you imagine TASTING something in your mouth, it will SURELY help kids remember.

Another thing I loved about her book were the little motivational "talks" in between the math stuff. For example, on p. 19 she asks, "Ugly Yourself Up?" Since you wouldn't make yourself to appear uglier than you really are, Danica argues, then why do the same as far as your smarts? She is truly encouraging girls to be smart, to study math, to do well in school and even venture into math careers!

It's really hard to put the book down, it's so delightfully written. How about remembering the order of operations by what Pandas Eat? And she goes further than the PEMDAS rule - Multiplication and Division are the main course, while Addition and Subtraction come at dessert.

For EVERY SINGLE topic she's come up with some kind of catchy illustration. For factoring, it's monkeys hanging off of trees. For the x (variable), it's a pearl bag, so 2x + 3 means "2 bags of pearls plus 3 loose pearls" (it's a GOOD illustration, I feel). For inequalities, she talks about how tall you'd want him to be? Perhaps h (his height) should be at least 2 inches taller but no more than 9... For isolating x in equations, she uses the analogy of gift wrapping and unwrapping. For exponents, there's the Ms. Exponent who's a high-powered executive. And so on.

You can see she's given each concept some thought, as to how to present it in a way that helps kids remember it.

She also helps us remember all these difficult math words (such as mean, median, mode, coefficient, variable etc.) with some clever mnemonics. I'll just tell you one. Mode is like "mood", whatever occurs most often, and she gives an example of a person who's iPod playlist contains mostly samba songs: "I guess she's in the mood for samba, huh?" Mode = mood.

The real power of the book is actually in these memorable analogies and in the very friendly tone she uses all through the book. She's talking to a friend, and it makes you to just keep reading!

Now, some might feel that it's all just nonsense. Will it really work to talk about Ms. Exponent being so powerful that all she has to do is say a word and things happen - you know, all she has to do is say "3", and 4 multiplies itself 3 times? Will it help students? Isn't it just silly to do math that way?

I feel it is truly powerful, because it has to do with our EMOTIONS. If it makes you laugh out of silliness, it creates an emotion while studying, and that helps you remember. It's a proven fact from brain research that we remember things better IF they're tied in with some strong emotion, be it fear, embarrassment, joy, surprise, etc.

You can also see from the feedback she's gotten that it IS working. Even math teachers have written to her how good her illustrative ideas are.

Another reason why this approach works is because she starts with the simplest stuff and in a friendly way "lures" you into the more difficult ideas. She basically "bends backwards" in order to not make math intimidating or scary - and that is one of her missions, as she has publicly declared.

I also liked the little "Watch Out!" boxes which contain alerts or explain common errors and misconceptions. Definitions come in "What's It Called?" boxes. Then in between the main text we also find "What's the Deal" boxes, Quick Notes, and Takeaway Tips, all decorated with smart and fashionable looking girl-figures.

Both books have some material on solving word problems. I feel Danica does an excellent job here and has really good common-sense advice AND great example solutions for this difficult topic. And I really liked chapter 12, The Art of Gift Wrapping/Solving Equations in Kiss My Math. It is one of the best "treatises" on solving linear equations that I've seen anywhere! The analogy of wrapping / unwrapping a gift is excellent.

Are the books just collections of tricks and shortcuts?

Some people have characterized Danica's Math Doesn't Suck book as simply a bag of tricks, concentrating on the "how-to" and not on the conceptual understanding. Well, it is written in a "how-to" style. It's like she's talking to you and coaching you through problems — and it's written so interestingly that it's hard to stop reading!

Sometimes she doesn't explain why things work the way they do, but most often she does include something along those lines - usually after the "how-to". So you can't say she's omitting the conceptual side (the why's). The books could use some more explanations on those lines, though.

Cons:

Now, I've checked around on the Internet, and it seems I may be a lone voice in the criticism that follows. However, I do not want to leave this part out so here goes.

You know, there are some of us (and I include myself in this) who don't feel we want our teenage daughters to start thinking about dating at such early ages as what her books audiences are. (Her books often refer to kissing, dating, having crushes, and so on.)

I'll give an example. She talks of integers as "mint-egers", which (like mints) taste like peppermint, spearmint, cinnamon and so on if they're positive. And they taste like vomit & dirt, etc. if they're negative. You combine (add) them in your mouth, and check which kind of taste wins. A great analogy! But right in the midst of several pages of discussing integers there's this one sentence: "These are good before a date, in case, you know, there's going to be kissing involved."

Another example: Chapter 9 in Kiss My Math is titled "Do You Like Him Like Him?" This catchy title is used as a mnemonic for the like terms in algebraic expressions — two terms such as 2x and 3x "get full-on crushes".


Now, she DOES also go against the popular culture in some ways. For example, she strongly encourages girls to study math, to be smart, to work at becoming smarter, not to feel that smart & pretty can't go together, not to underachieve and so on. The Kiss My Math book includes great testimonials from gals who overcame their struggles in math and are now "fabulously successful women".

I just wish she wouldn't go along with the "teen magazine culture" so much when it comes to crushes and boyfriends.

Recommendations

Teachers: I can heartily recommend these books for teachers, as they will be able to collect all kinds of ideas for their own teaching, and also learn from Danica's relaxed teaching style.

Parents and students: If you can keep from being influenced further into the boyfriend craze, go for it. If you can't, consider Dr. Math books (I've reviewed them as well). They are written in a somewhat similar, friendly, relaxed manner, though they don't contain nearly as catchy mnemonics and analogies as what Danica's do.

Review by Maria Miller

(The links below go to Amazon)
Math Doesn't Suck and Kiss My Math by Danica McKellar.

---

You may leave comments on this blogpost, but only intelligent discussion, no flaming, please!

Percent of change

When some quantity changes, such as a price or the amount of students, we can measure either the absolute change ("The price increased by $5" or "There were 93 less students this year"), or the percent change.

In percent change, we express WHAT PART of the original quantity the change was.

For example, if a gadget costs $44 and the price is increased by $5, we measure the percent change by first considering WHAT PART $5 is of $44. Of course the answer is easy: it is 5/44 or five forty-fourths parts.

To make it percent change, however, we need to express that part using hundredths and not 44th parts. this happens to be easy, too. As seen in my previous post, you COULD make a proportion to find out how many hundredths 5/44 is:

5/44 = x/100

To solve this, you simply go 5/44 x 100, which is easy enough to remember in itself. In fact, this is the rule often given: you compare the PART to the WHOLE using division (5/44), and multiply that by 100.

---

There were 568 students one year, and 480 the next year. By how may percent did the student population decrease?

You first calculate the absolute change, which is 568 − 480 = 88. Then we find what part of the original population is 88 (it is 88/568), and express that using hundredth parts (percents):

88/568 x 100 = 15.49%

The student population decreased by 15.49%.

---

Often we are given the opposite problem: we know the percent of change and the original situation, and are asked about the new situation.

The price was $4.55 and increased by 14.78%. What is the new price?

Here, we'd need to find the price increase, or the absolute change in price first. We know the percent part of the total (it is 14.78/100) and the total amount, so multiplying those we get the part as a dollar-amount: 14.78/100 x $4.55 = $0.67249. So this is the increase. To find the new price, add the increase to the original: $4.55 + $0.67249 = $5.22249 = $5.22.

Instead of multiplying by 14.78/100, it is far quicker to multiply by 0.1478 — or to change the percent-amount 14.78% to a decimal 0.1478 and multiply by it.

And, since in the end we need to add the original total, the whole calculation looks like this:

0.1478 × $4.55 + $4.55

Here, using distributive property we can make it look like this:

= $4.55 (0.1478 + 1) = $4.55(1.1478) = 1.1478 × $4.55

So it can all be done in one multiplication. Instead of multiplying by the decimal 0.1478, you add 1 to it before multiplying.

---

Then one more possible problem type is that you know the percent of change and the actual change amount (absolute change), and are asked the original and/or the new total.

The price increased by 13%, or by $10.14. What was the original price?

Let the original price be p. Then you can build an equation based on the idea that the price increase is 13% of p:

0.13p = $10.14   or   13/100p = $10.14

p = $10.14/0.13 = $78.

The new price would be found by adding.

---

Some other lessons to read are below:

Percent Of Change - Lesson and Problems

General Increase and Decrease Examples from Purplemath.com

Percent of change calculator - enter the original and the changed quantities, and it calculates the percent of change.

PERCENT INCREASE OR DECREASE lesson from TheMathPage.com

Some percent basics

The word "percent" means "per hundred", as if dividing by hundred — a hundredth part of something.



We treat some quantity (say 65 or $489 or 1.392 or anything) as "one whole". This "one whole" is then divided to hundred equal parts in our minds, and each such part is one percent of the whole.

If the "one whole" is 650 people, then 1% of it would be 6.5 people (if you have a practical application, you'd need to round such an answer to whole peoples of course).

If the "one whole" is $42, then 1% of it is $0.42. Also, 2% of it would be $0.84.

So to find 1% of something, divide by 100.
To find 24% or 8% or any other percentage, you can technically first find the 1%, then take that times 24 or 8 or whatever is your percentage.





For example:
To find 7% of $41.50, first go $41.50/100 to find 1% or 1/100 of $41.50, then multiply that by 7. But this is the same as (7/100) x $41.50, and 7/100 is 0.07 as a decimal. In most calculations, it is more practical to use decimals instead of this "divide by 100, then multiply" stuff.

So to find 7% of $41.50, I simply calculate 0.07 x $41.50 with a calculator.

To find 10% of something, you could first divide by 100 and then multiply by 10, but it's far quicker to simply divide by 10.

For example:
10% of 90 is 90/10 = 9.
10% of 250.6 is 25.06.

When you can find 10%, it's so easy to find 20%, 30%, 40%, etc., and 5% of anything just by using the 10% as a starting point.

For example:
20% of 52. First find 10% of 52 as 5.2, then double that: it is 10.4.

Example:
A gadget costs $48 and is discounted by 15%. What is the new price?

Imagine the price $48 is divided to 100 equal parts. Then you take a way 15 of those parts. That leaves 85 of those parts - what is the dollar amount that is left? Remember you're not taking away $15 but 15% of the total.

The student needs to realize that $48 is 100% - a "one whole", and 15 of those 100 parts will be taken away.

Solution:
10% of $48 would be $4.80.
5% of $48 would be $2.40 (half of 10%).
So 15% of $48 is $7.20. Subtract that from the orignal price to get the discount price of $40.80.
With a calculator, I'd go 0.85 x $48. (MAKE SURE YOU FIGURE OUT WHERE THE 0.85 COMES FROM!)

How many percent is it?

Of the class of 34 students, 12 are girls. How many percent of the class are girls?

Here, the "one whole" is 34, the whole class. The problem is, if that 34 "one whole" was divided to 100 parts, how many of those parts would we need to make 12 students?

Or, you could compare 34 people side-by-side with 100 "something". Imagine all those 34 people put head-to-toe so they form a long line, and those 12 girls are at the one end of that line. If you'd find 100 equal-size measuring units that would total exactly the same length as your people-line, how many of those measuring units would the 12 girls equal?

This easily leads to a percent proportion:

12/34 = x / 100

Solving x, you'd get
x = (12/34) × 100.

After you do this kind of proportion a few times, you notice that each time we just compare the part to the total using division, such as 12/34 in my example. So it's quite fast then to just write that directly, when solving "what percent" problems.

For example:
A $199 guitar was discounted by $40. How many percent discount was that?

Here, the "one whole" is the original (total) price, $199. It's simply asking how many percent is 40 of 199? Just calculate 40/199, and multiply the given decimal by 100 (which is easy to do mentally).

Hope this helps some. We'll tackle the percent of change next time.

So many percent more

Updated with an answer... see below

I'm continuing to catch up after vacation, and spotted a good discussion about problems with percent, at MathNotations. (via Let's Play Math blog).

Here's a problem to solve, first of all:

There are 20% more girls than boys in the senior class.
What percent of the seniors are girls?

The answer is NOT that 40% are boys and 60% are girls...

You see, let's say there were 40 boys and 60 girls, 100 students total. If there are 40 boys, then 20% more than that would be 40 + 4 + 4 = 48 girls and not 60!

Try solve it. I'll let you think a little before answering it myself. Don't just rush over to the Mathnotations blog either! Use your thinking caps! I've already given you a big hint!

Update:
You can easily solve this problem by taking any example number for the number of boys. Like I did above, if you have 40 boys, you'd need 48 girls and there'd be 88 students total. What percent of the seniors are girls then? It'd be 48/88 = 0.545454... ≈ 54.55%. And 45.45% are boys.

But the same works even if you choose that there'd be 10 boys, which then means that there are 12 girls, and then the percent of girls in the class is 12/22 * 100% = 54.55%.

NOTE that this problem includes two DIFFERENT "wholes". First of all, it says "There are 20% more girls than boys in the senior class." This is a comparison, and the total number of boys is the "one whole" or the "100%". The number of girls is 20% more, or 120% of the boys.

In algebra terms, if there are p boys, then there will be 1.2p girls.

Then the final question involves a totally different "one whole" or 100%: it asks how many percent of the seniors are girls.

So the group of seniors becomes the 100% or the "whole", and all percent calculations are based on that. Therefore one will then compare the number of girls to the total number of seniors.

In algebra terms, the final answer as a decimal is 1.2p/(p + 1.2p) = 1.2/2.2

See also Denise's post about searching for the 100% in percent problems.

Multiplying decimals

Someone asked,

how can you use models to multiply decimals?

Learning to multiply decimals, I feel, is built on students' previous understanding of multiplying whole numbers and fractions.

So models wouldn't necessarily be the focus, but instead relating decimals to fractions first, and learning from that.

Multiply a decimal by a whole number

Of course, when multiplying a decimal by a whole number, you could use the same models as for fractions: say you have a problem

2 × 0.34

You can use little hundredths cubes, or draw something that's divided to 100 parts.



BUT you can also just use fractions, and justify the calculation that way:

2 × 0.34 = 2 × 34/100 = 68/100 = 0.68.

OR you can explain it as repeated addition:

2 × 0.34 = 0.34 + 0.34 = 0.68.

I employ that idea in these lessons:

Multiply mentally decimals that have tenths and
Multiply decimals that have hundredths

Multiply a decimal by a decimal

When students are learning to multiply a decimal by a decimal, they're on 5th or 6th grade perhaps. One of the most obvious ways to teach this is to use fractions:

0.5	×	0.3	=
5 10	×	3 10	=	3 × 5 10 × 10	=	15 100

= 0.15.

With such calculations the student is made to notice that when both factors had one decimal digit (and thus had ten as denominator when written as fraction), then the result has two decimal digits (because you multiplied 10 × 10 to get 100 as denominator).

I do not know if there is any obvious pictorial model or manipulative that could be used to multiply a decimal by a decimal.

That is because it's no longer multiplying by a whole number.
It's no longer fitting the simple idea of "repeated addition".

It is instead best understood as 'taking part of' (just like multiplying a fraction by a fraction).

For example 0.1 × 45.9 is the same as taking 10th part of 45.9.

Or, 0.25 × 45.9 is taking 25/100 part of 45.9

And so on.

Understanding this is also crucial so that students can fathom how the answer number can sometimes be smaller than any of the factors... It's no longer adding where everything gets 'bigger' - it's taking a part of something.

See for example:

0.2 × 0.5 = 0.10

0.3 × 0.4 = 0.12

0.6 × 0.05 = 0.030

Here's one worksheet I've made that lets students practice the intricacies of decimal multiplication (a PDF file from my 5th grade worksheets collection).

Math Mammoth Grade 7 (prealgebra) worksheets available

Just as of today, I got online and available the new Math Mammoth Grade 7-A and 7-B worksheets collections.

These pre-algebra sheets contain problems for beginning algebra topics such as expressions, linear equations, and slope - plus your typical 7th grade math topics such as integers, fractions, decimals, geometry, statistics, and probability.

You will find the problems are very varying as these worksheets have been created one by one (not script made).

Go download and enjoy the free sample sheets! - including a Percent Fact Sheet.

Problem on proportional reasoning

I'm just going to solve here on the blog another math problem that was sent in to me... Hopefully it helps some of you to learn how to solve problems (you can let me know!)

A man travelled 8 miles in the second hour. This is 1/7 times more than during the first hour, and 1/4 times more than he travelled during the third hour. What is the total miles he covered in three hours?

In this problem, it is easy to get "deceived" and think that you'd just go 1/7 × 8 miles or something like that.

But think first; did the man cover MORE miles during the first hour than during the second hour?

Yes; it says plainly that the 8 miles was 1/7 times MORE than what he covered during the first hour.

Is that 8 miles a LOT more, or a LITTLE BIT more than what he traveled during the first hour?

It's 1/7 times more, so it's a little bit more.

So simply mark as x the miles covered during the first hour. Then,

8 miles = 1 1/7 x

It's not 1/7 x, but 1 1/7 x, or 8/7x. If you put down 1/7 x, you're finding the seventh part, but it was seventh part MORE.

So 8 miles = 8/7x

Multiply both sides by 7, and divide by 8:

x = 7 miles.

Then the last hour. Again the miles traveled during the second hour are more than the miles traveled during the third hour. If miles traveled during the third hour are y, then we get,

8 miles = 1 1/4 y

8 miles = 5/4 y

y = 32/5 miles, or 6 2/5 miles.

Total he covered 7 mi + 8 mi + 6 2/5 mi = 21 2/5 miles or 21.4 miles.

Teaching integers

Recently I answered a question about teaching integers on an email list, and decided to post all that on my site as well.

Also, I spent a few hours making downloadable fact sheets about all integer operations - these are free.

The sheets try to include quite a bit about "why" the rules for adding, subtracting, multiplying, dividing integers work.

Teaching integers
Enjoy!

---

Also I will answer here a question left on my site... there was no email address left so I couldn't answer via email.

how do you do this problems: elizabeth bought 3 1/3 pounds of tomatoes for $2.50. how much did she pay per pound?

Prices per pound are always given in [dollars per pound] or [dollars / pounds ].

This gives you the idea: you need to take the dollar amount and DIVIDE it by the pounds.

$2.50 ÷ 3 1/3 lb = $2.50 ÷ 10/3 lb = $2.50 × 3/10 = $7.50/10 = $0.75 per lb.