Part of the Whole
Below are two groups of pictures. What do you think each group has in common?
When you think you know what each group has in common, check your understanding by deciding which of the pictures below belong in group 1 and which belong in group 2.
(Don't keep reading down the page until you've tried thinking about this!)
(Don't keep reading down the page until you've tried thinking about this!)
Are you sure you're ready to read on?
Some answers are coming up....
Here's one way you might have thought about the groups and which pictures belong in them.
Group 1 shows only part of something and group 2 shows the whole thing. If that's the way you are thinking about them, you might sort the pictures this way: A, C, D, E, and H go in group 1, and B, F, and G go in group 2. (If you had a different way of thinking about the groups, you might have sorted the pictures differently! If you aren't sure how each picture shows a whole or a part, click here.)
There's something else that's special about group 1. It doesn't just show part of things, it shows a special kind of part. Can you tell what it is? Here are some more pictures to help you think about it. All of these pictures show part of something, but some of them belong in group 1 and some do not. Which ones belong in group 1? (Don't keep reading below the pictures until you've tried thinking about this on your own.)
Some answers are coming up....
Here's one way you might have thought about the groups and which pictures belong in them.
Group 1 shows only part of something and group 2 shows the whole thing. If that's the way you are thinking about them, you might sort the pictures this way: A, C, D, E, and H go in group 1, and B, F, and G go in group 2. (If you had a different way of thinking about the groups, you might have sorted the pictures differently! If you aren't sure how each picture shows a whole or a part, click here.)
There's something else that's special about group 1. It doesn't just show part of things, it shows a special kind of part. Can you tell what it is? Here are some more pictures to help you think about it. All of these pictures show part of something, but some of them belong in group 1 and some do not. Which ones belong in group 1? (Don't keep reading below the pictures until you've tried thinking about this on your own.)
If you've ever shared something fairly with another person, you know what one-half means. One-half means one of two parts that are the same size. Look at the pictures in group 1 again. Can you see how each one shows one-half? Click here if you want to check your answers about which of the pictures above show one-half and read some explanations.
One-half can be written as a fraction in other ways. Use the interactive model below to find some other fractions that are equal to one-half. Drag the orange dot to change the picture and find other fractions that also mean one-half.
The orange part of the rectangle will always be one-half of the whole rectangle.
The numbers in the boxes show the value of each box. To see the value of the whole, look at the number line or add up the values of all the boxes.
One-half can be written as a fraction in other ways. Use the interactive model below to find some other fractions that are equal to one-half. Drag the orange dot to change the picture and find other fractions that also mean one-half.
The orange part of the rectangle will always be one-half of the whole rectangle.
The numbers in the boxes show the value of each box. To see the value of the whole, look at the number line or add up the values of all the boxes.
Here are some challenges to try with the model above. For each one, try to predict what the answer will be and then check your answer by moving the orange dot in the model.
1. Find a fraction that is equal to 1/2 that has 7 as the top number.
2. Find a fraction that is equal to 1/2 that has 6 as the bottom number.
[Math vocabulary alert: The top number of a fraction is called the numerator. The bottom number is called the denominator.]
What pattern do you see in fractions that are equal to 1/2?
Here are some more fractions that are equal to 1/2. Do they fit your pattern?
1. Find a fraction that is equal to 1/2 that has 7 as the top number.
2. Find a fraction that is equal to 1/2 that has 6 as the bottom number.
[Math vocabulary alert: The top number of a fraction is called the numerator. The bottom number is called the denominator.]
What pattern do you see in fractions that are equal to 1/2?
Here are some more fractions that are equal to 1/2. Do they fit your pattern?
3. Try writing some other fractions that equal 1/2. What has to be true about a fraction for it to equal 1/2?
(After you've tried writing some of your own, you can click here for some more examples.)
(After you've tried writing some of your own, you can click here for some more examples.)
Other Ways of Saying One-Half
There are some other ways of talking about one-half.
One is 0.5 or 0.50. (You might see this meaning half of a dollar like this: $0.50)
Another way to say one-half is 50% (The % symbol means "percent.")
Just like you can think about half or 50% of a picture or a thing, you can also think about half or 50% of a number. (The whole thing or the whole number is called 100%.)
Use the interactive model below to explore what 50% looks like for different numbers. The picture shows a rectangle that is 50% or one-half orange. The numbers underneath tell you the values of 50% and 100%. Before you start playing with the model, it tells you that 50 is 50% of 100.
You can drag the black dot to change what number the whole rectangle stands for. What happens to 50%, when you make the whole bigger or smaller?
You can also drag the orange dot to make 50% bigger or smaller. What happens to the whole when you make 50% bigger or smaller?
Use the model to answer the following questions:
4. What is 50% of 110?
5. If 50% of a number is 65, what is the whole number?
6. What is a number that is less than 50% of 160?
7. If 50% of a number is more than 90, what could the number be?
Click here if you want to check your answers to the questions above.
4. What is 50% of 110?
5. If 50% of a number is 65, what is the whole number?
6. What is a number that is less than 50% of 160?
7. If 50% of a number is more than 90, what could the number be?
Click here if you want to check your answers to the questions above.
A Survey About Reading
Several towns were surveyed to see whether people preferred reading on paper or on a device like a phone or e-reader. Look at the table below.
8. Which towns had more than 50% of the people saying that they preferred paper? How do you know? (You can use the model above to help you think about this.)
8. Which towns had more than 50% of the people saying that they preferred paper? How do you know? (You can use the model above to help you think about this.)
Survey Results | |||
---|---|---|---|
Town | People surveyed | People who prefer paper | People who prefer a device |
Springfield | 100 | 40 | 60 |
Chester | 120 | 95 | 25 |
Greenville | 65 | 15 | 50 |
Arlington | 75 | 40 | 35 |
In Springfield, 40 out of 100 people surveyed said they prefer reading on paper, so the fraction of people who said they prefer paper was 40 out of 100 or 40/100.
9. Is this fraction more or less than 50%? How do you know?
10. Try writing fractions for the people who preferred paper in the other towns. Are those fractions more or less than 50%?
(If you want to check your work, click here for the answers.)
Here's some information about some other towns where the survey was conducted. The questions have more than one possible answer. Make sure you know why your answer works!
11. In Waverly, more than 50% of the people said they preferred reading on paper. If 80 people were
surveyed in Waverly, what is a possible number for the people who said they prefer paper?
12. In Riverside, 64 people said that they preferred reading on a device and this was more than 50% of the
people surveyed. How many people might have been surveyed in Riverside?
13. In Kingston, 80 people said they preferred reading on a device and this was less than 50% of the people surveyed. How many people might have been surveyed in Kingston?
Survey results are often shown in a circle graph. On the circle graph below, drag the slider to show what you think the survey results might look like from the different towns.
14. Can you make a graph that shows that more than 50% of the people prefer paper? If not, why not?
15. Can you make a graph that shows that more than 50% of the people prefer devices? If not, why not?
16. Can you make a graph that shows that more than 50% prefer paper and more than 50% prefer
devices. If not, why not?
9. Is this fraction more or less than 50%? How do you know?
10. Try writing fractions for the people who preferred paper in the other towns. Are those fractions more or less than 50%?
(If you want to check your work, click here for the answers.)
Here's some information about some other towns where the survey was conducted. The questions have more than one possible answer. Make sure you know why your answer works!
11. In Waverly, more than 50% of the people said they preferred reading on paper. If 80 people were
surveyed in Waverly, what is a possible number for the people who said they prefer paper?
12. In Riverside, 64 people said that they preferred reading on a device and this was more than 50% of the
people surveyed. How many people might have been surveyed in Riverside?
13. In Kingston, 80 people said they preferred reading on a device and this was less than 50% of the people surveyed. How many people might have been surveyed in Kingston?
Survey results are often shown in a circle graph. On the circle graph below, drag the slider to show what you think the survey results might look like from the different towns.
14. Can you make a graph that shows that more than 50% of the people prefer paper? If not, why not?
15. Can you make a graph that shows that more than 50% of the people prefer devices? If not, why not?
16. Can you make a graph that shows that more than 50% prefer paper and more than 50% prefer
devices. If not, why not?
(Click here to see some examples of graphs.)