Back to the on-screen lesson ·
Mixed practice on the fraction work from earlier units, with nothing new: unit fractions, fractions on a number line, equivalence, comparing, whole numbers as fractions and adding like fractions.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Nothing here is new. These are the fraction questions from earlier in the year, asked again now that time has passed and mixed up so that you have to work out what each one is asking before you can answer it. Remembering something yourself is what fixes it in place; being shown it again is not.
Earlier this year you met every idea in this lesson: unit fractions, fractions on a number line, equal fractions, comparing, whole numbers written as fractions, and adding fractions with the same bottom number. Nothing here is new. The questions are mixed up, so the first job each time is to work out which idea a question is asking about, and the second is to remember how that idea works.
| Term | What it means |
|---|---|
| Unit fraction | One equal part of a whole, with 1 on top, such as 1/5. |
| Numerator | The top number: how many equal parts are counted. |
| Denominator | The bottom number: how many equal parts make one whole. |
| Equivalent fractions | Different names for the same amount, such as 1/2 and 2/4. |
| Like fractions | Fractions with the same denominator, so their parts are the same size. |
All of the fraction ideas this year come from two facts. The bottom number tells you the size of one equal part of the whole. The top number counts how many of those parts you have. Naming, placing, comparing, renaming and adding fractions are all ways of using those two facts. When a question feels unfamiliar, say what each number in it means before you calculate.
Another way: routine
Name the whole, name one part, decide what the question wants you to do with the parts, and draw a quick bar or number line if it helps you see it.
In a mixed review, the hardest step is often deciding what is being asked. Read the whole question before writing anything. Look for clue words. A question that asks which part or how many equal parts is about unit fractions. A question with a line from 0 to 1, or jumps, is about the number line. A question that asks which fraction is the same amount, or gives a fraction with a blank, is about equivalent fractions.
A question with the words greater, less, or more is about comparing. A question that asks how many wholes, or how many parts are in several wholes, is about whole numbers as fractions. A question that puts two amounts together, or asks for a total, is about adding like fractions.
Saying the kind of question to yourself is not wasted time. It points you to the right method, and it stops you from using a rule where it does not belong, such as comparing top numbers when the bottom numbers are different. If you are not sure which kind a question is, look at what the answer should be: a single fraction, a point, a comparison, or a total.
A unit fraction is one equal part of a whole. If a whole is cut into 8 equal parts, one part is 1/8. Counting unit fractions makes other fractions: three of those parts is 3/8. The bottom number names the part; the top number counts parts.
On a number line, the whole is the distance from 0 to 1. Cutting it into 8 equal jumps makes each jump 1/8. To find 3/8, start at 0 and make three jumps. Remember to count jumps, not marks. The first jump lands on 1/8, not on 0.
The line keeps going past 1. With jumps of 1/4, four jumps reach 1, and the fifth reaches 5/4. A fraction whose top number is bigger than its bottom number is more than one whole. When placing such a fraction, find the whole number first and then count the jumps that are left. For 7/4, four jumps reach 1 and three more reach 7/4.
Equivalent fractions name the same amount in different-sized parts. Cut each half of a bar into two pieces and the half becomes two fourths: 1/2 = 2/4. Cutting every part into the same number of pieces multiplies both the top and the bottom number by that number. To find a missing number, ask what the bottom number was multiplied by, and multiply the top number the same way.
For example, 2/3 = ?/12. Three was multiplied by 4 to make 12, because each third was cut into four pieces. The two shaded thirds become 2 × 4 = 8 pieces, so 2/3 = 8/12. Check by asking whether the shaded amount changed. It did not; only the size of the pieces did.
Whole numbers have fraction names too. One whole cut into thirds is 3/3. Two wholes in thirds is 6/3, because each whole holds three thirds. To find how many parts are in several wholes, multiply the number of wholes by the parts in one whole. To find how many wholes a fraction makes, divide the top number by the bottom number.
To compare fractions with the same bottom number, compare the top numbers, because the parts are the same size: 5/8 > 3/8. To compare fractions with the same top number, compare the part sizes: the smaller bottom number has the bigger parts, so 3/4 > 3/8. When neither number matches, compare each fraction with one half.
To add fractions with the same bottom number, add the top numbers and keep the bottom number. Two sixths and three sixths make five sixths: 2/6 + 3/6 = 5/6. The parts are all sixths, so you are adding counts of the same unit. Do not add the bottom numbers. Writing 2/6 + 3/6 = 5/12 would claim that the parts got smaller just because you put them together, which a drawing shows is false.
Adding can carry you past one whole. 4/5 + 3/5 = 7/5, which is one whole and two fifths. Use the number line or a drawing to check: seven jumps of a fifth pass 1 after the fifth jump.
In a review there is no worked example right before each question, so checking matters more. After each answer, ask whether it is sensible. A unit fraction with a big bottom number should be a small amount. A fraction bigger than one whole should have a top number bigger than its bottom number. An equivalent fraction should name the same amount, not a bigger one.
Use a second method when you can. If you found an equivalent fraction by multiplying, check it with a quick drawing. If you added like fractions, count the jumps on a number line. If you compared two fractions, check with the halfway landmark. Two methods that agree give you good reason to trust the answer.
Do not erase a hesitant answer straight away. Mark it as not sure, finish the next question, and then come back. Often the second look is clearer. At the end, notice which kind of question felt slowest. That is the idea to practice next, and naming it exactly is more useful than deciding that fractions are hard.
When a fraction idea will not come back, a quick drawing often brings it. Draw a bar for the whole and cut it into the number of equal parts the bottom number names. Shade the parts the top number counts. The picture reminds you what each number means, and from there the method usually follows.
Different questions call for slightly different drawings. For comparing, draw two bars of the same length, one under the other, so the wholes match. For equivalent fractions, draw one bar and then cut each of its parts again, so you can see the same shading described in smaller pieces. For adding like fractions, shade the first amount, then shade the second amount in the next parts along, and count all the shaded parts.
Number lines are drawings too. They are especially good for fractions bigger than one, because a bar has to be repeated for every whole, but a line simply keeps going. Mark 0, 1 and 2 first, cut each whole into equal jumps, and count.
Keep drawings quick and honest. The parts do not need to be perfect, but they should look roughly equal, and the two bars in a comparison should be the same length. A careless drawing can suggest a wrong answer, so if a drawing surprises you, check it against the numbers before you trust it. A drawing is there to help you think, not to replace thinking.
A class orders pizzas for a party, and each pizza is cut into 8 equal slices. One slice is 1/8 of a pizza. A table of students eats 5 slices from one pizza, so they eat 5/8 of it, and 3/8 is left. Another table eats 3/8 of a different pizza. Both fractions are eighths of the same size of pizza, so the first table ate more: 5/8 > 3/8.
At the end the teacher combines the leftovers. One box has 3/8 of a pizza and another has 5/8. Adding like fractions, 3/8 + 5/8 = 8/8, which is one whole pizza. The leftovers fit into one box. If a third box held 2/8 more, the total would be 10/8, which is one whole pizza and 2/8, or one and one quarter pizzas.
Next time the class plans 3 pizzas. In eighths, that is 3 × 8 = 24 slices, so 3 = 24/8. With 24 students, each student can have exactly one slice. If the pizzas were cut into 6 slices each instead, there would be 18 slices, which is not enough. The fractions help the class decide how to order before anyone goes hungry. Fractions turned a guess into a plan.
Counting parts or adding fractions with the same bottom number changes only the top number. The parts do not change size, so the bottom number stays the same. Adding the bottom numbers is the most common mistake in a fraction review, and a quick drawing shows why it is wrong. Shade 1/4 of a bar and then another 1/4. Together they cover half the bar, which is 2/4. If the bottom numbers were added, the answer would be 2/8, which covers only a quarter of the bar: less than either amount you started with on its own. Putting two amounts together can never make less than one of them, so 2/8 must be wrong.
A second common mistake is to compare fractions by their bottom numbers alone, thinking 1/8 is more than 1/4 because 8 is more than 4. Ask how many people share the whole. Eight people sharing a pizza each get less than four people sharing the same pizza. A third mistake is to count the marks on a number line instead of the jumps, which makes fourths look like thirds. Each of these mistakes comes from forgetting what the bottom number means, so saying its meaning out loud is the best way to avoid all three.
Name the kind of question.
comparing
The question asks which is more.
Check the bottom numbers.
8 and 8
Both are eighths of the same size of pizza.
Compare the counts.
5 > 3
With same-size parts, more parts is more.
Write the comparison.
5/8 > 3/8
The open side faces the greater amount.
Check with one half.
5/8 > 1/2 > 3/8
Double 5 is 10 and double 3 is 6, against 8.
Name the kind of question.
equivalent fractions
A new name for the same amount.
Find the multiplier.
3 × 4 = 12
Each third is cut into four pieces.
Multiply the top number.
2 × 4 = 8
The two shaded thirds become eight pieces.
Write the equal fractions.
2/3 = 8/12
Same amount, smaller pieces.
Place it with twelfths.
8 jumps of 1/12
Count jumps from 0.
Check with one half.
8 × 2 = 16 > 12
So 8/12 sits right of the middle, as 2/3 does.
Name the kind of question.
adding like fractions
The leftovers are put together.
Check the bottom numbers.
8, 8 and 8
All the parts are eighths.
Add the top numbers.
3 + 5 + 2 = 10
Counts of the same part add.
Keep the bottom number.
10/8
The slices are still eighths.
Find the whole pizzas.
8/8 = 1
Eight eighths make one whole pizza.
Find what is left over.
10 − 8 = 2
Two slices remain after one whole pizza.
State the total.
10/8 = 1 and 2/8
One whole pizza and two eighths more.
Name the kind of question.
whole numbers as fractions
Parts in several wholes.
Count the parts in one whole.
4 fourths
Four fourths make one whole.
Multiply by the wholes.
Write the fraction.
A jump rope is cut into 8 equal pieces for a game. What fraction of the rope is one piece?
n/m of the rope
A tray of rice crispy treats is cut into 12 equal squares. A team takes 2 squares. What fraction of the tray does the team take?
n/m of the tray
A nature trail is one mile long, with a sign every 1/5 mile. A hiker stops at sign 3. Mark the hiker's place on the line from 0 to 1.
0 |——————————| 1
Mark the position with a cross, then write the value:
A movie lasts 3/2 hours. Mark its length on the line from 0 to 2 hours, marked in jumps of 1/2 hour.
0 |——————————| 2
Mark the position with a cross, then write the value:
Each part of 3/5 is cut into 3 equal pieces. Find the missing top number: 3/5 = ?/15.
Answer:
Write one half as a fraction with bottom number 8.
1/2 = n/8
Two teams paint the same fence. Team A paints 1/8 of it and Team B paints 6/8. Write the fraction for the team that painted more.
n/m of the fence
Each pizza is cut into 5 equal slices. How many slices are in 2 whole pizzas?
Answer:
One box holds 3/9 of a pie and another holds 4/9 of the same kind of pie. How much pie is there altogether?
n/m of a pie
Complete the steps that add 4/7 and 1/7.
Check the bottom numbers.
they match
The pieces are the same size.
Add the top numbers.
4 + 1 = top
Counts of the same piece add.
Keep the bottom number.
bottom number: bottom
Putting pieces together does not change their size.
One box holds 4/9 of a pie and another holds 3/9 of the same kind of pie. How much pie is there altogether?
n/m of a pie
You can still name a unit fraction, place one on a number line, tell whether two are equivalent, compare two with the same bottom number, and add them — mixed up, with nobody reminding you first. Whichever felt slowest is the one to go back to.
16. Your turn: how many fourths are in 3 wholes, step 3
3 × 4 = 12
Each of the three wholes holds four fourths.
16. Your turn: how many fourths are in 3 wholes, step 4
3 = 12/4
Twelve fourths make three wholes.