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Counting pieces of the same size, and why the bottom number does not change.
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.
In this lesson you add fractions that have the same bottom number. When the pieces are all the same size, adding is just counting them: two eighths and three eighths are five eighths. The bottom number stays as it is, and understanding why — the size of a piece has not changed, only how many you have — is what stops the two numbers being added together later on.
You can name fractions, explain what each written number means, and recognize a whole as all its equal parts. This lesson extends those ideas to addition and subtraction with a common denominator. It is a retained Grade 4 extension, built on Grade 3 unit-fraction reasoning. Keep a strip or a number line available. You should be able to explain why the pieces being combined have the same size before using the written calculation. A matching denominator matters because it names a matching unit.
| Term | What it means |
|---|---|
| Common denominator | The same denominator in fractions being considered together. |
| Sum | The total when amounts are added. |
| Difference | The amount left or the distance between quantities after subtraction. |
| Addend | One of the quantities being added. |
| Unit fraction | One equal part of a whole, used as the counting unit. |
| Non-overlapping parts | Parts that do not count any of the same area or amount twice. |
Two eighths and three eighths make five eighths. All the pieces are eighths of the same whole, so addition combines their counts: 2 + 3 = 5. It does not cut every piece again. The denominator therefore stays eight, giving 2/8 + 3/8 = 5/8. This is like combining two centimetres and three centimetres to get five centimetres. The number changes, but the unit does not. A drawing or a count of equal jumps should tell the same story as the symbols.
Another way: steps
Check that the whole and unit size match. Add or subtract the counts. Keep the unit name. Check the result with the model and an inverse calculation.
Imagine a paper strip divided into eight equal parts. Colour two parts blue and three different parts yellow. There are five coloured parts altogether, each one eighth of the strip. The total coloured fraction is 5/8. The colours help distinguish the two addends, but they do not change the size of a piece. Counting all eight parts again confirms that the whole still has eight equal parts after the colouring.
The word 'different' matters. If a blue part is painted yellow on top, counting it once as blue and again as yellow would count the same area twice. To add fractions of distinct covered areas, the areas must not overlap. If the question instead counts the amount of paint used, two coats might both count. Decide what quantity the story asks for before writing an addition.
The whole matters too. One fourth of a large sheet and one fourth of a small sheet are not equal-area pieces merely because both are called fourths. To combine physical areas in a single fraction, choose a common reference whole or use an area unit. In the practice here, fraction addends refer to the same whole unless a question explicitly asks you to inspect that condition.
First speak the unit-fraction calculation: two eighths plus three eighths equals five eighths. Then write the same calculation with fractions: 2/8 + 3/8 = (2 + 3)/8 = 5/8. The parentheses show that the numerator is the combined count. They do not change the denominator. You are using one kind of piece throughout the calculation.
A number line gives another model. Divide the unit from zero to one into eight equal intervals. Start at zero, move two intervals, then move three more. You land at the fifth interval endpoint. Each jump remains one eighth of a unit. The first jump of the second move starts where the first move ended; it does not restart at zero.
Now check the sum against the addends. With positive fractions, adding another amount must increase the total. Five eighths is greater than both two eighths and three eighths, which fits the story. If a proposed sum is smaller than an addend, inspect the units and the operation. A magnitude check catches mistakes that a remembered rule might miss.
Someone adds 1/4 and 1/4 and writes 2/8. The top counts were added, but the bottom counts were also added. Test the claim with a strip. One fourth and another fourth cover two of the four equal parts, which is half the strip. Two eighths cover only one fourth. The proposed answer has not grown when another positive amount was added.
The error changes the unit during the calculation. Eighths are half the size of fourths. If you choose to repartition into eighths, each selected fourth becomes two selected eighths. Then 2/8 + 2/8 = 4/8, which is still half the strip. A valid repartition changes every relevant count consistently. Merely adding the denominators does not do that.
This is why an explanation should name the common unit. Saying 'the denominator never changes' is too broad: later work can deliberately replace fractions with equivalent names. The precise statement here is that adding counts of the same unit leaves that unit unchanged.
A route is divided into eight equal lengths. You walk one eighth, then two more eighths, then three more eighths. Count 1 + 2 + 3 = 6 eighths in total. You can combine the first two counts before the third, or combine the last two first. Both ways count all six equal intervals exactly once.
The total is 6/8 of the route. Two eighths remain because eight eighths make the complete route and eight minus six equals two. Check by adding the completed and remaining parts: 6/8 + 2/8 = 8/8 = 1. This whole check is especially helpful in stories about a task being completed.
Adding can also reach or pass one whole. Three fourths plus one fourth is four fourths, exactly one. Three fourths plus two fourths is five fourths, one whole and another fourth. The rule does not stop working when the numerator reaches the denominator. Continue counting the same unit, then regroup complete wholes if that description is useful.
Subtraction with matching denominators removes a count of the same unit. From seven eighths, remove three eighths. Four eighths remain because 7 − 3 = 4. Write 7/8 − 3/8 = 4/8. The remaining pieces are still eighths; removing some does not make the others larger. Check by adding back: 4/8 + 3/8 = 7/8.
A missing-addend question uses the same relationship. In 2/6 + ?/6 = 5/6, ask how many sixths take the count from two to five. The difference is three, so the missing amount is 3/6. Do not answer just three if the question asks for the fraction: three wholes would be a much larger amount.
When a story asks what remains of one whole, write the whole using the same denominator before subtracting. For sixths, one is 6/6. If four sixths are used, 6/6 − 4/6 = 2/6 remains. This notation makes every unit visible and avoids subtracting four from the whole-number symbol one.
The word 'more' does not always tell you to add. Suppose a question says one child has walked five eighths of a route and another has walked three eighths, then asks how much more the first child has walked. The unknown is the gap between two positions, so subtract: 5/8 − 3/8 = 2/8. The same fractions in a story about two successive sections walked by one child would be added. The relationship, not one word, chooses the operation.
Sketch both situations. For the comparison, draw two arrows starting at zero and mark the distance between their endpoints. For the combined journey, draw the second arrow starting at the first arrow's endpoint. These pictures lead to different equations because they describe different questions. Before calculating, say whether you are finding a combined amount, an amount left, a gap or a missing part. Then check that your answer carries the unit fraction used throughout the story.
A result may have more than one correct name. Four eighths and one half have the same value. If the question asks you to keep the original piece size, write 4/8 to show the count of eighths. If it asks for simplest form, regroup pairs or larger equal groups to obtain 1/2. Neither notation changes the result's value.
Simplifying is a separate step from addition. First combine matching units correctly. Then choose an equivalent name if needed. For 2/6 + 2/6, the immediate sum is 4/6. Pairing all sixths turns them into thirds, and the four selected sixths become two selected thirds. Thus 4/6 = 2/3. Explain the regrouping rather than treating crossed-out numbers as the reason for equality.
Do not use this lesson's procedure to add unlike units directly. One half and one third are not two copies of a single unit fraction. They require a common partition before their counts can be combined. Recognizing that the units do not match is the appropriate decision here; a general unlike-denominator algorithm belongs to later work.
A class makes one long banner divided into eight equal-area panels. One group decorates two panels in the morning. A second group decorates three different panels in the afternoon. The groups have completed 2/8 and 3/8 of the same banner. Since the panels are equal and no panel is counted twice, they have completed 5/8 of the banner altogether.
The whole banner is 8/8. Subtract the completed part: 8/8 − 5/8 = 3/8. Three panels remain. Check the plan by accounting for every panel: two morning panels, three afternoon panels and three unfinished panels make eight panels. The denominator stays eight in every fraction because the banner and its partition remain fixed.
These area fractions do not tell us how much time the remaining work will take. One panel may need more detailed decoration than another. Equal areas do not guarantee equal working times. They also do not prove that the groups used equal amounts of paint. More coats or different materials could change consumption. The supported conclusion is about completed and unfinished area. To plan time or materials, ask for those measurements separately. A useful fraction calculation is precise about the quantity it describes and does not silently turn an area share into a claim about every other part of the project.
Adding denominators changes the part size without a valid repartition. Subtracting a fraction from a whole requires naming that whole in the same unit. Adding areas requires checking that they do not overlap and that their reference whole matches. Use the inverse operation and a model to test the amount, then follow the requested form of the final answer.
Check the reference whole.
One strip
Both collections use this same strip as a unit.
Identify the common unit.
Sixths
Both denominators are six.
Combine the counts.
1 + 3 = 4
There are four selected sixth-pieces.
Write the sum.
1/6 + 3/6 = 4/6
The piece size has not changed.
Check by removing one addend.
4/6 − 3/6 = 1/6
The inverse recovers the other addend.
Read the completed shares.
1/8 and 4/8
They cover distinct panels of the same banner.
Add their counts.
1 + 4 = 5
Each count uses eighths.
Name the completed share.
5/8
Five equal panels are complete.
Name the whole in eighths.
1 = 8/8
Eight panels make the entire banner.
Find the remaining share.
8/8 − 5/8 = 3/8
Remove completed panels from the whole.
Check all panels.
5/8 + 3/8 = 8/8
Completed and remaining areas cover the banner exactly.
Choose the common unit.
One strip, partitioned into fourths
Both amounts use identical strips.
Read the addends.
3/4 + 2/4
Each fraction counts fourth-pieces.
Combine selected counts.
3 + 2 = 5
All selected pieces are equal in size.
Write the exact sum.
5/4
Five fourths have been counted.
Separate a complete whole.
5/4 = 4/4 + 1/4
Four fourths can be grouped together.
Describe the result.
One whole and one fourth
The extra fourth must not be discarded.
Check by subtracting the second addend.
5/4 − 2/4 = 3/4
The original first amount is recovered.
Read the relationship.
3/8 + ?/8 = 7/8
The missing amount completes the total.
Find the missing count.
7 − 3 = 4
Subtract the known count from the total count.
Keep the unit.
Check by addition.
Work out 2/9 + 6/9 and write it as {{n}}/{{m}}, keeping denominator 9.
n/m
1/8 of a route is walked, then 3/8 more. Complete the total and remaining shares in the original unit.
Combine the completed counts.
1 + 3 = done
Both fractions count the same length unit.
Remove that count from a whole.
8 − done = left
The complete route contains 8 such units.
Retain the common denominator.
Both shares use denominator 8.
Walking changes the count, not the size of the parts.
Ana walks 4/12 of the way to school, then 1/12 more, then 2/12 more. What fraction of the way is that? Write it as {{n}}/{{m}}, keeping denominator 12.
n/m
Why does $\frac{5}{11} + \frac{1}{11}$ keep 11 as its denominator?
Work out 6/10 + 1/10 and write it as {{n}}/{{m}}, keeping denominator 10.
n/m
Ana walks 1/8 of the way to school, then 1/8 more, then 4/8 more. What fraction of the way is that? Write it as {{n}}/{{m}}, keeping denominator 8.
n/m
A wall has 6 equal panels to paint. Nia paints 2/6 of it before lunch and another, non-overlapping 3/6 after. What fraction of the wall is still unpainted? Write it as {{n}}/{{m}}, keeping denominator 6.
n/m
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Two walkers start at the same point on the same route. One has walked 4/11 of it and the other 2/11. How much farther has the first walked? Keep the original denominator.
n/d
You can add two fractions with the same bottom number. Without looking: what is 2/7 plus 3/7, and why does the 7 stay the same?
17. Find a missing number of eighths, step 3
Missing amount = 4/8
The missing count is a count of eighths.
17. Find a missing number of eighths, step 4
3/8 + 4/8 = 7/8
The completed equation matches the given total.