Math-Fractions

Adding 3 tenths + 4 tenths is exactly the same as adding 3 dollars + 4 dollars or 3 hours + 4 hours. What does that mean?

"Adding 3 tenths + 4 tenths is exactly the same as adding 3 dollars + 4 dollars or 3 hours + 4 hours" refers to the idea that when adding you must add things that are the same type or size. For decimals, for example, rather than adding tenths with hundredths you add tenths with tenths. When adding fractions this means you need to add together the same size parts. So, in order to add the numerators of two fractions together, the fractions have to represent the same size parts, which means they need to have the same denominator. This is needed because the denominator represents what size parts of the whole you have. In general, you would also add dollars with dollars, cents with cents, hours with hours, and x's with y's, to name a few examples

three areas important to understanding fractions

(1) Reasoning with Fraction Models (2) Benchmark Fractions (common fractions used to compare other fractions to) and Reasoning with Number Lines (3) Fractions as Indicators of a Size.

When are unlike denominators addressed?

5th grade children need to use understanding of equivalence

Developing Images for Fractions by Marking Number Lines

Asking students to partition and mark number lines to show fractional parts can support visualization of the size of various fractional units as indicated by the denominator.

Developing Images for Fractions by Placing Them on a Number Line

Asking students to place fractions on a number line and explain their reasoning can provide another opportunity to engage students in reasoning about the denominator as an indicator of size. As students become proficient with locating fractions on a number line, they can develop a sense for how big various fractions are based on reasoning about size as suggested by the denominator.

What are children learning about fractions in 3rd grade

Develop understanding of fractions as numbers. (unit fractions, which is when the numerator is 1) Grade 3 expectations in this domain are limited to fractions with denominators of 2, 3, 4, 6 and 8.

When students understand these 3 factors....

If these three areas are developed well, students will be able to figure out how to add and subtract fractions by using visual models. From their work with visual models, you can help them make sense of what they symbolism means.

Models

Models help students visualize fractions and support an understanding of their size and location

Definition of denominator: How many equal parts there are in all. What should be changed?

The definition of a denominator is missing a key piece of information, which means it should be slightly altered. Besides representing how many equal parts the whole is divided into, the denominator is also an indicator of size. This means the denominators represents what size parts of the whole you have. For the fraction 2/8, for example, the 8 means that the whole is split into eight equal parts and each part is in the size of an eighth (1/8). (Depending on the numerator, I would have "x" number of these eighths. In this example, it I would have two eighths) This is the reason that when adding and subtracting fractions it can only be done with like denominators, or when the fractions have the same sized parts. The definition of denominator should be, "The denominator represents the number of parts the whole is divide into and the size of each of these parts."

Then, explain the important mathematical idea that allows this to happen.

The mathematical idea that allows this to happen, at least for decimals and wholes, is the idea of place value. Place value refers to the value of a digit given its position in a number. When you add decimals, for example, the location of the digit tells the value of it, which may be tenths, hundredths, or thousands. This allows you to tell what digits should be added to what digit based on their value. In general though, the mathematical idea that allows this to happen is the idea that likes can only be added to or subtracted from one another. This applies to fractions, decimals, money, time, and variables, to name a few. This ties into the idea of why focusing on equivalence of fractions with students is so important. Without knowing how to find equivalent fractions, students won't be able to use this skill when adding and subtracting fractions with unlike denominators. (For example, if I'm adding one fourth and one eighth, I can just add these together and say I have 2 eighths. It doesn't make sense. Each fraction represents a different sized piece, which means I need find a fraction equivalent, in this case, to one fourth. I know one fourth split in half is the same as two eighths. The numerators of two eighths and one eighth can then be added together because represented the same sized pieces.) When children understand this idea of equivalence it will help them solve problems, while also affirming the mathematical idea that only likes can be added to or subtraction from likes. Along with that, it's important to stress that models and visuals are key to understanding this mathematical idea that only like sizes or types can be added or subtracted from one another. By using models, children can better understand what fractions, dollars, decimals, and money represent. For example, by understanding what 3/6 and 1/6 look like when splitting two wholes of the same size into sixths students could see that these fractions could be added together because they both represent the same sized pieces. Models and visuals are so important in helping children understand these concepts and lays the foundation for working with and understanding symbols.

How is this similar to adding whole and decimals?

You have to add things that are the same type or size. You add tens with tens, ones with ones, tenths with tenths, hundredths with hundredths. With decimals and whole number we as student to learn their place value chart and name the place values. With fractions, we use pictures to show same size parts. Just like with whole numbers and decimals you add 10ths with 10ths, 100ths with 100ths, 4ths with 4ths, 6ths with 6ths, etc... AND (!!!) it doesn't stop there. With money we add dollars with dollars and cents with cents. If you have a quarter and a nickel you express that in cents—25 cents and 5 cents. In algebra you add x with x and y with y. So 3x + 2y + 4x + 2y is found by adding the "x"s with "x"s and the "y"s with "y"s. Whenever you add or subtract, the units need to be the same size if you want to find out how many you have.

When the size of the parts are the same you can

add them Ex: if both fractions are eighths you can add the numeators

Denomintor

an indicator of the size of a unit of measure Ex: any fraction with a denominator of five is a quantity composed with some number of 1/5-size parts of a whole (along with the number of equal parts in a whole)

By asking questions such as, "What does the 6 in the fraction tell you?" or "How many sixths does it take to make 5/6?, students are encouraged to see the denominator?"

as the number of parts the whole is partitioned into and that the size of each of the five parts in 5/6 is one sixth.

how to add and subtract fractions with similar denominators

can start with word problem Draw a picture and cut it into equal sized parts based on the fraction In the video this was eighths Label each portion of pizza was eaten by each person Shade that area. By doing this you are adding the fractions

make sure to ask questions that ....

deepen student understanding of fractions Ex: How far is 5/6 from zero? How far is 5/6 from one?

Focus of fraction in fourth grade

developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers

One activity to do this is by

draw a large number line with markings at 0, 1/2, and 1 on the board. Students can be given sticky notes with various fractions written on them, and asked to go to the board to determine approximately where to place the fraction on the number line. can connect to fraction strips to support their ideas For example, asking where the fraction 1/4 would be located on the number line and how you know this can lead to a discussion about how fourths are smaller than halves. S

What does this mean about symbols?

even though symbols may be introduced and worked with in third grade always connect symbolism with visuals. don't take away visuals very quickly

Marking and labeling a number line can support reasoning about how

far one fraction is from another fraction

Asking students to count as they label a number line partitioned into sixths (0/6, 1/6, 2/6, . . . ) helps to....

focus students' attention on the idea that the denominator tells you that the whole is partitioned into six equal-size parts same idea applies to other fractions

make sure to stratefically choose the order in which

fractions are presented to students

The denominators is an

indicator of size six of the parts in/make up the whole

It's important to have students

label the fractions they use on number lines label each notch, if given, that makes up the portion of the whole you have

What is the most common way to subtract mixed numbers?

make mixed numbers into improper fractions then subtract

In our own professors experience she found even though students were suppose to be able to work with symbols....

many of them really didn't understand what a fraction represented. Visuals were important!

Students need to do a lot of problems with ? to learn to add fractions with common denominators.

models

How to subtract using improper fractions? (from video she sent in email)

multiple whole by denominator add number and numerator to make the new numerator repeat for other fraction and simply them if needed find common denominator s multiply denominator to get the denominator of both fractions the same multiple top as way of this same number then subtract can then switch back to mixed number

To find equivlent fractions in the one video, it showed

she represented fraction on number line and labeled in she drew another number line making sure to line the whole numbers up (if they don't line up it's comparing differed sized fractions) she then partitioned each fraction in smaller fractions , determined what fraction was now represent compared the two to find equivalent fractions

What do most textbooks and curricula do though?

show a picture or two in a lesson but then shift to symbols.

Focus of fractions in 5th grade

students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. visual models are still important

Besides using fraction strips it's important students

students be asked to draw a lot (lots and lots and lots) of pictures. This helps them develop a visual understanding of fractions. By drawing lots of different fractions students develop a strong visual understanding of the size of fractions such as—what are 8ths and 4ths—and how are 4ths and 8ths related.

To find equivalent fractions....

students can use number line or fraction strips ask students to think about how 8ths and 4ths are related or how 3rds and 6ths are related. How does the size of an 8th compare to the size of a 4th. Having students work on tasks where they draw number lines is really important.

What does fluency refer to?

students will become proficient with adding and subtracting with symbols but fluency also means that students can explain and justify what they are doing with models too

fraction number sense

the ability to reason about the size of different fractional units

numerator

the number of equal parts you have the number of the fractional size you have (ex: 2/5 I have two fifths)

Developing Images for Fractions by Folding Paper Strips

used to help students visualize fractions and their size in relation to other fractions

When children work with fractions in third grade it's very, very....

visual children use -Number lines -visual fraction models (like the fraction strips you folded)

In fourth grade students continue to use what?

visual models students use to be able to justify and explain their reasoning

So in 3rd and 4th be sure to work on

visual models for finding equivalent fractions before and along with procedures for renaming fractions. Fractions and number lines are great but you do eventually have to teach a rule to make equivalent fraction.

It can be easy to start with a

word problem This helps students see from the beginning how fractions are used in life. And think about this too—when you are out in the world, life does not throw fraction flashcards at you to solve. You have to make sense of real-world contexts. (Most textbooks though, most operations are taught with symbols and models, and a few word problems, or the word problems come last)