Week 4

Tuesday April 16th

Tuesday we reviewed expanded notation in decimal form

Take the following problems for example:

12.57~ 10+2+4/10+7/100

3.056~ 3+0/10+5/100+6/100

When writing in expanded notation if there's a zero in any position, such as the zero in the problem above it's not necessary to write it out, but for me personally it makes it a lot easier to remember which place value the numbers have.

We then went on to expanded exponential notation:

342.5~ (3x10^2)+(4x10^1)+(2x10^0)+(5x10^-1)

The final thing we reviewed was finding percents of numbers, we learned that one of the easiest ways to do so is with a number line, a strategy I'd never used before for this type of problem. It really did turn out be easy to understand in a more conceptual way.

For example: 60% of 30

By drawing the above number line you can see that adding 50% and 10% of 30 would be like adding 15 and 3 and your answer would be 18.

Thursday April 18th

We learned how to solve ratio story problems like the following:

Molly bought six heads of cabbage for $9.30, Willie goes to the same store and needs 22 heads of cabbage. How much will it cost him?

The easiest way to solve a problem like this is to set up a ratio table.

Cost, Number of Heads of Cabbage

4.65   3

9.30   6

18.60  12

23.25   15

32.55   21

34.10   22

By solving the problem this way you're just using the knowledge you already have to divide the amount until it's a manageable number. 

For the rest of the day we finished our final homework assignment :)

Hope this post was helpful!

Week 3

Tuesday April 9th

Today we discussed how to add and subtract fractions without common denominators. One way, the way most of us use first, is to find a common denominator and add or subtract the numbers like normal. This can be confusing though, even to us, so it may not be the best way to teach children.
Take the following for example:

7 1/3-4 3/4

My first thought when seeing this problem was to change both fractions to a denominator of 12, resulting in 7 4/12 and 4 9/12. This can get confusing though since 4 is less than 9, so you then have to subtract 5/12 from a whole, which is where most people make mistakes.

A more simple way, that is also more visually understandable, would be to use a number line.

In order to find the answer to the problem you add the numbers along the top (1/4+2+1/3) which comes to 2 7/12.

We then moved on to multiplication and division of fractions. When discussing division of fractions, for example 4/6 divided by 3/4. You would write it as 3/4 of 4/6, and as most of us from elementary school can remember of=multiply.

We did a couple of example problems, and on Thursday we had a more in depth lesson on the division of fractions.

Thursday April 11th

As said above today we went more into division of fractions. The rule we've all been taught is to invert (flip) the second fraction (the dividend) and multiply it by the first number (the divisor). But none of us really know why we do this.

There are different ways of dividing fractions that actually allow us to conceptually understand what we're doing.

Partitioning: take the dividend and split into groups.

Repeated Subtraction: How many of the divisor go into the dividend.

For instance:
3 divided by 1/2...most people will answer this question with 1.5 (myself included)

The correct answer is actually 6. Because the problem isn't telling you to divide 6 in half, it's asking how many 1/2 groups are in 6.

We then went on to discuss an easier way to divide fractions, if the two fractions have common denominators (or you convert them so they do) you can just focus on the top numbers (the numerators) and divide those.

Take for example:

12/25 divided by 4/25

Since the denominators are both 25 you can just focus on the numerators and do 12 divided by 4, which equals three, which is your answer! Talk about easy.
Some people were confused by this. Why can you just ignore the bottom number?
Well if you don't ignore the bottom number you can see you end up with the same solution....

12/25 divided by 4/25 you would end up with 12÷4 over 25÷25...well 12÷4=3 and 25÷25=1 so you would end up with 3/1 which equals three. So knowing that the denominators equate to one, you can simply skip this step and just work with the top numbers.

We ended this class with some discussion  of converting fractions to decimals and percents.

Hope this was helpful!

Week Two

Tuesday 4/2/13

We discussed fraction sense, for instance:

Write/draw everything you can that represents the fraction 3/4

Both of the above would be examples.

We then discussed multiple rules that are necessary in the understanding of fractions.

Spatial Relationships: (The pictures above are both examples of spatial relationships)
Having a picture of the number, including where it lies on a number line.

One/Two(Units) More and Less:
If I have 5/4 what is 1/4 more or less? What is 2/4 more or less?

Benchmarks:
1,1/2,1...is it above or below 1/2. How far away from 1?
ex. is 3.4 more or less than 1/2? It's more. How much more? 1/4.

Part-Part Whole:
Knowing 3/4 can be into parts; 1/2+1/4=3/4 or 1/4+1/4+1/4=3/4

These rules applied to the fraction 7/5:

Spatial Relationship:

The above would show spatially where 7/5 is on a number line (the line in red)

One or Two More/Less:
 one more is 8/5 or 1 3/5 one less is 6/5 or 1 1/5

Part-Part-Whole
 3/5+4/5=7/5 or 1/5+1/5+1/5+1/5+1/5+1/5+1/5=7/5

Thursday 4/4/13

On Thursday we discussed how to add and subtract fractions with different denominators. We discussed how it makes a lot more sense in many problems not to just find the simplest common denominator and add fractions that way, but instead to find a common denominator that represents something in the problem and actually makes sense to children (and college students).

Take this problem for example:
If I ran for 1/3 of an hour and then walked for 1/4 of an hour. How much time would that be? What fraction of an hour is that?

My first instinct would be to convert 1/4 and 1/3 to fractions with a common denominator so I could just add them (resulting in 4/12+3/12=7/12). This was very easy, but results in a fraction that isn't easily understood in relation to the problem.

So what should I do instead?
Well since this is a problem that has to do with time, there are 60 minutes in an hour. I could instead convert 7/12 into a fraction with a denominator of 60. To do this I  multiply both top and bottom numbers by 5. My resulting fraction is then 35/60. Not only is this fraction easy to understand in relation to the problem, but it also tell me how much time I was active for, 35 minutes.

This lesson taught that kids need to understand that when solving a problem the whole unit is more important that just the lowest common denominator.
Take the problem above for example, leaving the answer with correct units (60) instead of simplifying tells you more about the problem.

Hope this was helpful!

Week One

Tuesday March 26th~

What is a Unit?

On Tuesday in class we discussed how one whole unit doesn't necessarily have to be "one."
For instance:
   If you eat 1 and 1/2 grapefruits every morning and you have a total of 5 grapefruits and you want to know how many days they will last, you wouldn't find the answer to this problem by dividing 5 by 1. Instead you would find the answer by dividing 5 by 1 and 1/2 since that is the whole unit you are working with.

1 and 1/2 grapefruits

This skill is applicable when you are shown part of a picture and told it's a fraction of a whole...
For instance:
If you're told this is 3/4 of the whole, what would the whole be?

3/4 of the whole

This would be the whole: (not at scale)

Next we discussed a basic understanding of fractions and how important it is to explain them to kids correctly. Most of us were taught that if you see a fraction such as 3/5 you should think of it as 3 out of 5. Which is all fine until you get to a fraction like 4/3 which will completely confuse a child because they can't conceptually understand that from the 4 out of 3 explanation.

This is how you should instead explain it:
 3/5 should instead be explained as Three 1/5  pieces. That way when you get to fractions such as 4/3 you can explain it as Four 1/3 pieces. This method easier for children to comprehend and unlike the simplistic way is actually something they can apply to more difficult problems as they progress in school.

Three 1/5 pieces would be three of the above sections...

Big Ideas: Partitioning and Iterating

Partitioning: Splitting the whole into equal parts.

So if you're told the above rectangle is a whole unit and to find 1/4 you would "partition"  it into 4 equal pieces

(Image not to scale) your result would be something like this, with the unshaded piece representing 1/4

Iterating: Consistently repeating a unit to build a whole

So if you were told the above triangle is 1/3 and you were to find a whole unit you would iterate the triangle three times.

 

(not to scale) You can see how 3 triangles iterated results in a trapezoid shape.

Thursday March 28th~

Saying Fractions:
 We reviewed as discussed above how to properly say fractions, for maximum comprehension, (3/4 would be three 1/4 pieces)

We then discussed the definition of numerator and denominator:
Numerator: The NUMBER of pieces.
Denominator: The DENOMINATION (size) of the pieces.

For the rest of the class period we did activities that discussed whole units and fractions and their meaning.

Hope this post was helpful! :)