A woman without hands - inspiring!

An inspiring slideshow about Jessica Cox - a woman born without hands. She even learned to fly!

Jessica cox english.pps

A new template

Hope you like it! I just chose one that Blogger offers. They sure have improved their templates in the last few years because when I signed up with Blogger years ago, the templates were kind of lame... much better now.

How to help a student with a fraction word problem

Today, my daughter had to tackle this word problem in her 5th grade Math Mammoth (lesson Multiply Mixed Numbers):

The square on the right measures 10 in. × 10 in. and
the rectangle inside it measures 6 7/8 in. × 3 1/2 in.
How many square inches is the colored area?

This requires the student to

• multiply mixed numbers, 
• subtract mixed numbers, 
• understand about area, and 
• understand how to find the "colored" area by subtraction of areas. 

So it is a multi-step word problem.

She didn't understand it, she said. My first "help" was this:
"Let's say we change those fractions to whole numbers 6 and 3. Can you mark those in the image? Would you be able to solve the problem now?"

The strategy I used is:
If you can't solve the problem at hand, change it and make it easier. Then try to solve the easier problem.

She was able to mark 6 and 3 on the sides of the rectangle (that is inside the square). But she said she couldn't solve it. She said it's not possible to find the area of the colored area!


Then I asked her, "Is there anything you CAN find? Is there anything you CAN solve using this information?"

And this is another great strategy in solving any problem (whether math or not): If you can't find the answer to the question in the problem, solve what you CAN solve. That might lead you to find the answer to your question somewhere along the way!

I said, "Well, we CAN find the area of the square. It is 100 square inches. We CAN find the area of the rectangle inside it."

THEN immediately after I said that, she saw it: "OHH! SUBTRACT!" And on she went to multiply the mixed numbers in the problem. So the story had a happy ending!

---

Here's the complete solution:

First multiply the mixed numbers 6 7/8 and 3 1/2 to find the area of the rectangle. Keep in mind they need to be changed into fractions before multiplying.

6 7/8 × 3 1/2 = 55/8 × 7/2
= 385 / 16 square inches

Change this into a mixed number. Here one needs to use long division to divide 385 ÷ 16 = 24 R1. This tells us the whole number part is 24. The remainder,1, tells us how many 16th parts are "left over."

So, 385 / 16 = 24 1/16 square inches. So this is the area of the rectangle.

Now, the area of the surrounding square is simply 10 in x 10 in. = 100 square inches.

And lastly, the area of the colored area is found by subtracting 100 − 24 1/16 = 75 15/16 square inches.

Decimals videos

Here are some of my recent additions to Math Mammoth Youtube channel.
 - videos about decimal arithmetic.

Add and subtract decimals
I explain the main principle in adding or subtracting decimals: we can add or subtract "as if" there was no decimal point IF the decimals have the same kind of parts--either tenths, hundredths, or thousandths. Many students have a misconception of thinking of the "part" after the decimal point as "plain numbers." Such students will calculate 0.7 + 0.05 = 0.12, which is wrong, and I explain why in the video.






Multiply decimals by whole numbers
I explain how to multiply decimals by whole numbers: think of your decimal as so many "tenths", "hundredths", or "thousandths", and simply multiply as if there was no decimal point. Compare to multiplying so many "apples". For example, 5 x 0.06 is five copies of six "hundredths". Multiply 5 x 6 = 30. The answer has to be 30 hundredths (hundredths corresponding to apples), or 0.30, which simplifies to 0.3.




Divide decimals using mental math
I explain two basic situations where you can use mental math to divide decimals: 1) Think of "stuff" (which is tenths, hundredths, or thousandths) shared evenly between so many people; OR 2) Think how many times the divisor fits into the dividend.





Long division with decimals
When the dividend is a decimal, and the divisor is a whole number, long division is easy: just divide as if there was no decimal point, and then put a decimal point in the answer in the same place as it is in the dividend. I also show an example where we add decimal zeros to the dividend, in order to get an even division. Lastly I show how the fraction 3/7 is converted into a decimal using long division.

Green tea word problem

Photo credit: leojmelsrub

Ahmed sent me in this kind of word problem:


A tea producer want to market mixed green tea leaves at $14 per pound. how many pounds of high mountain green tea leaves worth $20 per pound must be mixed with 90 pounds of regular green tea leaved worth $10 per pound?

I have solve many problems like this one on my blog, but it never hurts to solve some more. This can be solved with algebra, using a chart. I've done that before for similar problems... so if you are reading this, and you feel a bit "rusty" in this area, try to make the chart yourself first, before you read further!

For the chart, we also need to choose a variable or several. In this case it is easy:  the unknown is obviously what is asked, or the amount of high mountain green tea. Note also that the cost is always the price per pound times the amount.

mountain green   regular green  the mixture
 tea leaves  tea leaves
--------------------------------------------------------------------------
amount |   x   90   90 + x
--------------------------------------------------------------------------
cost   |  20x   $900   14(90 + x)
--------------------------------------------------------------------------

The chart is ready. Its purpose is to help us write an EQUATION of some sort which will solve x.

So where can we find something equals something? It comes from the cost. The COST of mountain green tea + the COST of regular green tea = COST of the mixture.

20x + 900 = 14(90 + x)

20x + 900 = 1260 + 14x

6x = 360

x = 60 

Now, let's check. That's always the last step in solving equations.

We need 60 pounds of mountain green tea leaves mixed with 90 pounds of regular green tea leaves. The mixture will weigh 150 pounds. The cost of mountain green tea leaves will be $1200, the cost of regular green tea will be $900, and the total cost will be $2100. Calculating cost per pound: $2100 / 150 lb = $14 per pound, so it checks.

Math vocabulary resource

Would you like to help children with their math vocabulary? SpellingCity has built a resource to address this: Math vocabulary spelling lists.

If you don't know SpellingCity, no matter what kind of spelling list you use, you can always practice the words in many ways: either just simple practice AND with several different games: MatchIt Sentences, Which Word (find which word correctly completes the sentence), sentence unscramble, hang mouse, word search, word unscramble, etc. Can't even list them all.

So it is definitely a very comprehensive math vocabulary resource!

Two problems about fractional parts

I have 2 questions on fractions which I can't solve.

There were 3/5 as many adults as children on a bus. At the next bus stop, 6 adults and 6 children boarded the bus. As a result, there were 2/3 as many adults as children on the bus. How many people were on the bus at first?

A solution using bar diagrams (Singapore style):

children |----|----|----|----|----|

adults   |----|----|----|

Then we have:

children |----|----|----|----|----| +6

adults   |----|----|----| +6

Here look at the difference of children and adults. There are two "blocks" more children than adults. We know the number of adults is 2/3 of the number of children... therefore the DIFFERENCE of two blocks must be 1/3 of the children.

Now look:

children |----|----|----|----|----| +6

Two of those blocks is 1/3 of the total... the other two blocks is another 1/3 of the total... so |----| +6 or one block and 6 must be 1/3 of the total.
This means the +6 must be one block. Or, one block = 6. This now solves the problem, because originally we had 8 "blocks" of people, or 48 people.


Solution with algebra:
Let c be the number of children at first. There were (3/5)c adults at first.

Then 6 more adults and 6 more children come in, so now we have c + 6 children, and (3/5)c + 6 adults.

Now, the amount of adults = (2/3) of the amount of children.

(3/5)c + 6 = (2/3)(c + 6)

As this equation has fractions, let's multiply both sides by 15 to start.

9c + 90 = 10(c + 6)

9c + 90 = 10c + 60

30 = c

There were 30 children and 18 adults at first, or 48 total people.

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2/5 of the counters in a box were red and the rest were blue. After putting another 48 blue counters into a box, 3/4 of the counters were blue. How many counters were in the box at first?

A solution using bar diagrams (Singapore style):

blue |----|----|----|

red  |----|----|

Then:

blue |----|----|----| + 48

red  |----|----| 

...and now 3/4 of the counters are blue. So, |----|----| (the red ones) or two blocks is 1/4 of the counters. Therefore there are a total of 8 blocks of counters. but I drew five blocks and 48. So, 48 must be worth three "blocks", or one block is 16.

At first we had five "blocks" of counters, or 5 x 16 = 80 counters.

CHECK:
80 counters... 32 red and 48 blue.
Add 48 blue. Now we have 32 red and 96 blue, a total of 128.
Reds are 32/128 = 8/32 = 1/4. That checks.

A solution using algebra:
At first, we have (2/5)c counters red and (3/5)c counters blue. ONce we hadd 48 blue counters, we have (2/5)c red counters and (3/5)c + 48 blue counters, and c + 48 total counters. It says now 3/4 of the counters are blue. We write that as an equation:

(3/4) of all counters = number of blue counters

(3/4)(c + 48) = (3/5)c + 48 Multiply this by 20.

15(c + 48) = 12c + 960

15c + 720 = 12c + 960

3c = 240

c = 80

There were originally 80 counters.