How Fractions, Decimals and Percents Go Together…Like PB & J How Fractions, Decimals and Percents Go Together…Like PB & J! Charline Brown

How do our kids view fractions? Most students see parts-to-whole relationships only. Explain the numerator as “number you have” and the denominator as “number you have in all”? Students have limited concrete understanding of fractions.

How do our kids NEED to view fractions? Fractions represent parts-to-whole relationships Fractions are division problems Fractions can show ratio of unrelated things Fractions are numbers with unique locations on a number line.

Fraction Machines: The routine that helps develop concrete understanding of fractions. Students must fully understand the whole. When the whole is divided into separate equal pieces, fractions are formed and can be named.

Fraction Machines: I am going to take the whole and put it through this marvelous machine. It is called the halves machine. What do you expect will happen?

Fraction Machines: Can two pieces that look like this come out of the halves machine? Why or why not? What does half mean?  

Fraction Machines:   I am going to put this whole into the halves machine. How many pieces do I expect? 2 equal pieces What is the name of each piece? Each piece is called 1 2 How many pieces does it take to make a whole? 2 pieces make a whole

Fraction Machines:   one – half 1 out of 2 1 2 A whole that is placed into the halves machine will emerge to look like this. I will label the pieces.

Fraction Machines:   one – half 1 out of 2 1 2 Remember that each piece has the name of “half” because it takes 2 pieces to make a whole. What is a name? A fraction gets its name from the number of pieces it takes to make one whole. The denominator is the name of a fraction.

Fraction Machines:   I am going to put this whole into the thirds machine. How many pieces do I expect? 3 equal pieces What is the name of each piece? Each piece is called 1 3 How many pieces does it take to make a whole? 3 pieces make a whole

Fraction Machines:   one - third 1 out of 3 1 3 A whole that is placed into the thirds machine will emerge to look like this. I will label the pieces.

Fraction Machines:   I am going to put this whole into the fourths machine. How many pieces do I expect? 4 equal pieces What is the name of each piece? Each piece is called 1 4 How many pieces does it take to make a whole? 4 pieces make a whole

Fourths machine is broken! Fraction Machines: Uh, Oh!! Fourths machine is broken! Can you think of a way to make fourths using the machines that we have already used?

Fraction Machines:   First put the whole through the halves machine. Then take those 2 pieces and put them through the halves machine. 1 2 of 1 2 = 1 4

Fraction Machines:   one - fourth 1 4 1 2 of 1 2 1 out of 4 A whole that is placed into the fourths machine will emerge to look like this. I will label the pieces.

Fraction Machines:   I am going to put this whole into the fifths machine. How many pieces do I expect? 5 equal pieces What is the name of each piece? Each piece is called 1 5 How many pieces does it take to make a whole? 5 pieces make a whole

Fraction Machines:   one - fifth   1 5 1 out of 5    1 5  one - fifth A whole that is placed into the fifths machine will emerge to look like this. I will label the pieces.

Fraction Machines:   I am going to put this whole into the sixths machine. How many pieces do I expect? 6 equal pieces What is the name of each piece? Each piece is called 1 6 How many pieces does it take to make a whole? 6 pieces make a whole

Sixths machine is broken! Fraction Machines: Uh, Oh!! Sixths machine is broken! Can you think of a way to make sixths using the machines that we have already used?

Fraction Machines         First put the whole through the thirds machine. Then take those pieces and put them through the halves machine. 1 2 of 1 3 = 1 6

Fraction Machines       First put the whole through the halves machine. Then take those 2 pieces and put them through the thirds machine. 1 3 of 1 2 = 1 6

Fraction Machines:   one - sixth   1 6   1 2 of 1 3   1 3 of 1 2 1 out of 6 A whole that is placed into the sixths machine will emerge to look like this. I will label the pieces.

Fraction Machines: Let’s think about the fraction machines we have used. 1 2 , halves machine 1 3 , thirds machine 1 5 , fifths machine We were able to use some combination of these machines to make fourths and sixths. We are in a default budget and can only afford to purchase the machines that we absolutely NEED. Can you establish a pattern about the machines that need to be purchased?

Fraction Machines: These are the fractions made in this activity. Think about the machines that you need. 𝑥 2 𝑥 3 𝑥 4 𝑥 5 𝑥 6 𝑥 7 𝑥 8 𝑥 9 𝑥 10

Fraction Machines:   I am going to put this whole into the sevenths machine. How many pieces do I expect? 7 equal pieces What is the name of each piece? Each piece is called 1 7 How many pieces does it take to make a whole? 7 pieces make a whole

Fraction Machines:   one - seventh   1 7 1 out of 7    1 7 A whole that is placed into the sevenths machine will emerge to look like this. I will label the pieces.

Fraction Machines:   I am going to put this whole into the eighths machine. How many pieces do I expect? 8 equal pieces What is the name of each piece? Each piece is called 1 8 How many pieces does it take to make a whole? 8 pieces make a whole

Fraction Machines: 1 2 of 1 2 of 1 2 = 1 8               First put the whole through the halves machine. Then take those 2 pieces and put them through the halves machine. Then take those 4 pieces and put them through the halves machine. 1 2 of 1 2 of 1 2 = 1 8

Fraction Machines:           First put the whole through the fourths machine. Then take those 4 pieces and put them through the halves machine. 1 2 of 1 4 = 1 8

Fraction Machines:       First put the whole through the halves machine. Then take those 2 pieces and put them through the fourths machine. 1 4 of 1 2 = 1 8

Fraction Machines:   one - eighth   1 8 1 out of 8    1 2 of 1 4     1 4 of 1 2 1 2 of 1 2 of 1 2 A whole that is placed into the eighths machine will emerge to look like this. I will label the pieces.

Fraction Machines:   I am going to put this whole into the ninths machine. How many pieces do I expect? 9 equal pieces What is the name of each piece? Each piece is called 1 9 How many pieces does it take to make a whole? 9 pieces make a whole

Fraction Machines:         First put the whole through the thirds machine. Then take those 3 pieces and put them through the thirds machine. 1 3 of 1 3 = 1 9

Fraction Machines:   one - ninth   1 9 1 out of 9   1 3 of 1 3    1 9 A whole that is placed into the ninths machine will emerge to look like this. I will label the pieces.

Fraction Machines:   I am going to put this whole into the tenths machine. How many pieces do I expect? 10 equal pieces What is the name of each piece? Each piece is called 1 10 How many pieces does it take to make a whole? 10 pieces make a whole

Fraction Machines:             First put the whole through the fifths machine. Then take those 5 pieces and put them through the halves machine. 1 2 of 1 5 = 1 10

Fraction Machines:       First put the whole through the halves machine. Then take those 2 pieces and put them through the fifths machine. 1 5 of 1 2 = 1 10

Fraction Machines:   one - tenth   1 10 1 out of 10   1 2 of 1 5 1 5 of 1 2 A whole that is placed into the tenths machine will emerge to look like this. I will label the pieces.

Fractions Students have built a concrete understanding of fractions that show parts-to-whole relationships. They know that the denominator is the name. The name is the number of pieces to make a whole. Now they are ready for operations with fractions.

Fractions Can you add 1 pencil plus 2 pencils? Can you add 4 pencils plus 3 pencils? Can you subtract 3 pencils from 6 pencils? Can you subtract 4 pencils from 5 pencils? Can you add 2 pencils plus 3 notebooks? Can you take 1 pencil from 2 notebooks?

Fractions What is the difference between these examples? You can’t add or subtract the pencils and notebooks because they are not the same. They have different names. What implications does this have for addition and subtraction of fractions?

Fractions Can you add 1 sixth plus 2 sixths? Can you add 2 sixths plus 3 sixths? Can you subtract 3 sixths from 5 sixths? Can you subtract 2 sixths from 3 sixths? Can you add 3 sixths plus 1 fourth? Can you subtract 4 sixths from 3 fourths?

Fractions What is the difference between these examples? When the fractions have the same name (are like and have the same denominator), you can simply add or subtract them. When the fractions do not have the same name (are unlike with different denominators), you must change the name or names to add or subtract them.

Fractions Consider the following: Can you add 3 sixths plus 1 fourth? I can’t add them because their names are not the same. I must rename the fraction. In “fraction speak” I must make an equivalent fraction or fractions, changing the number of pieces it takes to make a whole.

Fractions 3 6 renamed is 6 12   1 4 renamed is 3 12   6 12 + 3 12 = 9 12

Fractions Consider the following: Can you subtract 4 sixths from 3 fourths? I can’t subtract them because their names are not the same. I must rename the fraction by making an equivalent fraction.

Fractions 3 4 renamed is 9 12   4 6 renamed is 8 12   9 12 − 8 12 = 1 12

Fractions What is 2 cookies times three? Can you think of a number story that can demonstrate this problem? The answer is 6 cookies. What is 4 tickets times 5? The answer is 20 tickets. What implications does this have for multiplication of fractions times a whole number?

Fractions Consider the following: What is 2 eighths times 3 ? Remember that eighths is a name (just like cookies). The name doesn’t change in this case just because you multiply it. You simply multiply the number of eighths times 3. 2 8 ∗ 3= 6 8 -- The name (denominator) doesn’t change.

Fractions Think back to the Fraction Machines. Remember that 1 2 𝑜𝑓 1 2 = 1 4 . In math, what does “of” mean? What is 3 groups of 5? What about 2 of 8? You naturally applied an operation! What was it?

Fractions In math, “of” implies multiplication. Therefore, 1 2 𝑜𝑓 1 2 = 1 2 ∗ 1 2 = 1 4 The name changed! We also discovered that 1 2 ∗ 1 3 = 1 6 ? What are we doing to the numerators and denominators to get the result we know to be true? Aha, you have established a rule!

Consider the area model of multiplication: 3 times 4 equals 12 Fractions Consider the area model of multiplication: 3 times 4 equals 12  

Fractions Consider the area model of multiplication: 4 6 * 1 3 = 4 18 Does your rule still work?  

Fractions It seems that my rule works! I can multiply fractions by simply multiplying the numerators to get the numerator in the answer. Next, I multiply the denominators to get the denominator in the answer.

Fractions Consider this problem: For dinner, my mom ordered 2 pizzas. 3 of us shared those pizzas equally. How much pizza did each person eat?

Use the Problem Solving Structure to answer. Fractions Use the Problem Solving Structure to answer.

2 pizzas shared by 3 people. Fractions 2 pizzas shared by 3 people.  Lisa    Marco  Emily  Lisa    Marco  Emily Each person’s share was 2 3 of a whole pizza. Those numbers are familiar! Fractions represent division problems!

Fractions But what about dividing fractions by whole numbers and other fractions? Consider this problem. You and two friends are sharing one pizza. How many pieces are you dividing the pizza into?

Fractions You are taking the 1 pizza and dividing it by 3. What is your share of this pizza? Your share of the pizza is 1 3 . So let me get this right! Dividing something by 3 (÷3) is the same as having 1 3 of that something?

Fractions And remember that in math, “of” implies multiplication. Therefore, I can solve 1 3 ÷ 3 by looking at it differently. Instead, I can think about it as 1 3 𝑜𝑓 1 3 . I much prefer multiplication of fractions! And, this explains the reciprocal (invert and multiply).

Decimals Now we have a pretty good understanding of fractions and how they behave with operations. Remember, the denominator of a fraction represents its name. The name is the number of parts that it takes to make one whole. Think about place value. The places have names, too.

Decimals I know whole number place value and I understand how each place relates to the places around it. I know that it takes 10 ones to make 1 ten, 10 tens to make 1 hundred, 10 hundreds to make 1 thousand and so on. The decimal lets the math observer know that they have crossed from whole number to partial number territory.

Decimals 1111.111 In this number, I know that each digit is 10 times smaller than the digit to its left. My last whole number is 1, and the first digit to its right (past the decimal) is 10 times smaller. 1 ÷ 10 is 1 10 (Fractions are division problems, after all!) The name of that place is tenths.

Decimals 1111.111 1 10 of 1 10 is 1 100 The digit to the right of the tenths is hundredths. 1 10 of 1 100 is 1 1000 The digit to the right of the hundredths is thousandths.

Decimals The decimal place value names are tenths, hundredths, or thousandths. Fractions go by many more names than decimals! The easiest way to convert a fraction into a decimal is to try to rename it. When I rename a fraction, I change the number of pieces to make one whole. That is an equivalent fraction.

Decimals What fraction names can easily be renamed to tenths or hundredths? 𝑥 2 𝑥 4 𝑥 5 𝑥 20 𝑥 25 𝑥 50

But what about fractions that don’t have those names? Decimals But what about fractions that don’t have those names? Remember, all fractions are division problems! How do I find the decimal equivalent for 6 7 ?

6 7 represents the division problem 6 ÷7. Decimals 6 7 represents the division problem 6 ÷7.

Let’s try a few more examples! Decimals Let’s try a few more examples!

Decimals You can turn any fraction into its decimal equivalent by renaming it as a decimal place value name or by solving its represented division problem.

Percentages Now that you understand fractions and decimals a bit better, you are ready for percentages! Think about where and when you have seen percentages in the world around you.

Percentages are used in: Grading Discounts in shopping Interest rates for loans or credit cards To describe increases or decreases

Percentages You can turn any fraction or decimal number into its percentage equivalent. Let’s think about spelling tests. Many weekly spelling lists are comprised of 20 words.

You are THE spelling Rock Star of the class! You didn’t miss one word. Percentages Let’s say that you practiced really hard this week on your spelling words. You are THE spelling Rock Star of the class! You didn’t miss one word. What is the fraction that represents your number correct out of the total number of words?

That is 1.00 – One whole test correct! Percentages That’s right! You got 20 20 ! That is 1.00 – One whole test correct! Your grade is 100% because you spelled ALL of the words correctly.

Now consider that you didn’t study at all! Percentages Now consider that you didn’t study at all! You only spelled half of the words correctly. Some fractions representing your score are 10 20 , 50 100 , 5 10 ,and 1 2 .

Percentages It takes 100 percentage points to make 1 whole. You always want to try to rename your fraction as hundredths to report it as a percentage. Remember the fraction names that can easily be turned into decimal place value names. 𝑥 2 𝑥 4 𝑥 5 𝑥 20 𝑥 25 𝑥 50

Percentages There is an easy way to find the decimal and percentage equivalents which reflect my score. 10 20 = 50 100 = .50 = 50%

Let’s try a few possible spelling test results. Percentages Let’s try a few possible spelling test results. 17 20 = 85 100 = .85 = 85% 9 20 = 45 100 = .45 = 45% 15 20 = 75 100 = .75 = 75%

Percentages But, what happens if I leave early and only had the chance to spell 18 words? My score was 13 out of 18, or 13 18 That is 13 ÷ 18.

Percentages 13 18 = 0.722 = 72%

Fractions, Decimals, and Percents Games that keep you learning! Decention - Math Playground Jeopardy - Fractions, Decimals and Percents Fruit Shoot - Fractions to Decimals Fraction and Percents Comparisons