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Generating equal fractions, simplest form, and why multiplying both numbers is allowed.
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.
You have met equal fractions once. This lesson goes further: generating a whole family of them from one fraction, finding the simplest member of that family, and seeing why multiplying the top and the bottom by the same number leaves the amount alone. It is the same idea as before, met a second time and further in — which is how an idea moves from recognised to owned.
You have shown equal fractions with bars and with points on a number line. You know that repartitioning every piece equally can change both written numbers without changing value. In this lesson, use those ideas to examine a proposed equality, generate a missing name and explain why an attractive but incorrect answer fails. Keep a strip of paper or a sketch available. An arithmetic rule should agree with a model. When the two disagree, the disagreement is useful: it tells you to inspect the whole, the partition or the way you counted the selected pieces.
| Term | What it means |
|---|---|
| Value | The number or amount represented, which stays fixed in an equivalent fraction. |
| Common factor | A whole-number factor shared by two numbers. |
| Equivalent name | Another fraction that represents the same value. |
| Simplest form | A fraction whose numerator and denominator share no whole-number factor greater than one. |
| Counterexample | A case that shows a proposed general rule does not always work. |
| Common whole | The same unit amount used when comparing physical fraction models. |
A true equation such as 2/3 = 4/6 can be explained rather than merely recognized. Each third is split into two sixths, so the selected two thirds become four sixths. Both the selected amount and the whole remain unchanged. That explanation links the two numbers in the fraction to quantities in a model.
This lesson asks you to work in both directions. Sometimes you start with larger pieces and split them; sometimes you start with smaller pieces and regroup them. Sometimes a classmate shows two equal-looking regions but uses different whole shapes. Then you must decide whether the drawing supports the stated fraction equality. An answer should include the changed counts and the condition that makes their comparison meaningful.
The main Grade 3 cases use halves, thirds, fourths, sixths and eighths. Some existing practice develops a later simplification procedure with other denominators. Use the same equal-piece reasoning there, but do not let a large-number rule replace your understanding of the small models.
Another way: steps
Name the whole; identify the selected amount; describe the split or regrouping; change both counts together; verify equal size or equal number-line position.
Start with a whole strip cut into halves, one half selected. Split every half in two to make fourths. The selected half now contains two fourths. Split every fourth in two again to make eighths. The same selected amount now contains four eighths. You have generated 1/2, 2/4 and 4/8 without changing the selected region at either step.
Notice that the second step starts from fourths, not from halves. Two selected fourths each become two eighths, so there are four selected eighths. All four fourths each become two eighths, so there are eight parts in the whole. If you count only the cuts in the selected region, you can lose track of the denominator.
You can record an equivalence family in a table with columns for selected parts, all parts and the size of one part. The selected count increases as each piece gets smaller. The fraction value remains one half throughout. Reading all three columns makes the change visible. Reading only the selected-count column would wrongly suggest that the amount grew from one to two to four. Fractions require keeping the part count connected to the part size.
In 3/4 = ?/8, the denominator doubles. Each selected fourth therefore becomes two eighths, so three selected fourths become six eighths. The missing numerator is six. In 3/4 = 6/?, the numerator tells you the same repartition: three pieces became six, so every piece was split in two. The denominator must become eight.
Now reverse the direction. In 6/8 = ?/4, the whole's eight pieces are paired into four larger pieces. Pair the selected six too, giving three selected pairs. The missing numerator is three. You are not taking away three pieces of the selected amount. You are counting the same amount with larger units, so fewer units are needed.
A blank may appear in any position, and the equal sign can be read in either direction. Do not assume every task means multiply the numbers on the left. First identify the known change. If parts become smaller, counts increase; if complete parts are regrouped, counts decrease. Then apply the same transformation to the other count. Finally rewrite the complete equation and check it with a model or by reversing the transformation.
Consider six eighths. To regroup into fourths, pair the eighths. Eight pieces in the whole make four pairs; six selected pieces make three pairs. The result is 3/4. Both counts were divided by two. Dividing only the selected count would give 3/8, which selects less of the whole. Dividing only the whole count would give 6/4, which describes more than one whole.
For four sixths, groups of two give two thirds. Try groups of three instead: the whole six pieces form two groups, but the selected four do not form a whole number of three-piece groups. That regrouping will not produce whole-number counts for both positions in this exercise. Choose a factor that fits both counts, not just the denominator.
The extended simplest-form procedure repeats this idea until no further whole-number regrouping is possible. The simplest name is useful, but the earlier name was not wrong. Six eighths and three fourths are equally accurate. A direction asking for simplest form specifies the name to write; it does not change the amount or imply that other equivalent names describe different numbers. Explain that distinction when checking a partner's work.
A classmate proposes adding the same number to the numerator and denominator. Test it with one half. Adding one gives two thirds. Draw equal-length strips: one split into two equal pieces with one selected, and one split into three equal pieces with two selected. The second selected length extends beyond the first. One counterexample shows that this proposed rule does not always preserve value.
Why does multiplying both counts work when adding does not? Multiplication describes splitting every original piece in the same way. Adding one extra selected piece and one extra piece to the whole does not generally describe such a repartition. The selected share can change. Always connect the operation to what happens to the pieces.
A second false shortcut is that any fractions with even numerators must be equivalent. Compare two fourths and two eighths of the same whole. Both count two pieces, but eighths are smaller than fourths, so the amounts differ. A pattern in one written number cannot decide equivalence. The numerator and denominator work together. A counterexample helps you identify the missing condition rather than merely calling an answer wrong.
The fraction numbers 1/2 and 2/4 are equal. But a picture of half a large poster and two fourths of a small poster does not show equal areas of paper. Each fraction refers to its own whole. For a fair physical comparison, first make the posters the same size or measure their selected areas in a shared unit.
Suppose one rectangle covers eight unit squares and another covers four. Half the first covers four squares, while two fourths of the second cover two squares. The fraction numbers are equal but the selected areas are not, because the wholes differ. If both rectangles instead cover eight squares, half and two fourths each cover four squares.
When presented with an unfamiliar drawing, say which question you are answering. Are the fraction numbers equal? Are the selected physical amounts equal? The first is about positions on a common number line; the second also depends on the whole amounts. Clear language prevents an apparent contradiction. A larger shaded region does not automatically represent a larger fraction of its own whole.
A chain such as 1/2 = 2/4 = 4/8 makes several claims at once. Check each neighboring pair. The first split doubles both counts, and the second split doubles both counts again. Since each move preserves value, every name in the chain represents the same number.
Now inspect 1/2 = 2/4 = 3/8. The first equality is correct. The second is not: doubling fourths into eighths should also double two selected parts into four. An early correct step does not make all later steps correct. Point to the first place where the relationship changed and repair that step.
Use a second representation when you are unsure. A common number line puts the correct labels at the same point. An aligned strip shows the same selected endpoint. Grouping small pieces back into larger ones reverses a valid split. These checks should agree. A complete explanation says which transformation was made, why it applies to both counts and what stayed unchanged. The machine can check entered numbers; a listener must judge whether your original explanation connects the model to those numbers.
Two classes are painting equal-sized display boards. One class divides its board into four equal regions and finishes three. The other divides its board into eight equal regions and finishes six. The second class has completed more regions by count, but each region is smaller. Pairing its six completed eighths makes three fourths, so both classes have painted the same area. Each has one fourth of its board still to paint.
The caretaker wants one consistent reporting system. If every board is reported in eighths, the first class should write six eighths completed and two eighths remaining. Those two amounts together make eight eighths, the entire board. A completed-plus-remaining check is useful because it examines both parts of the whole, not just the selected part.
This comparison relies on equal board areas and equal partitions. If one board is larger, matching fraction progress does not mean matching painted area. It also does not establish which class worked faster, because the time spent was not supplied. Report only what the evidence supports: equal fractions of equal boards correspond to equal completed areas. Ask for board dimensions or time records if a different question needs those measurements. Mathematical accuracy includes respecting the limits of the given information.
Changing both written numbers is not enough: the change must preserve their relationship. Adding one to each usually does not. Multiplying both by the same positive whole number corresponds to a uniform split and does preserve it. When regrouping, the common factor must fit both counts. When comparing physical pieces, the whole amounts also matter. An explanation that names those conditions is stronger than one that merely says to do the same thing to both numbers.
Identify the original name.
1/4
One of four equal parts is selected.
Find the whole's new part count.
4 × 2 = 8
Each fourth is split in two.
Find the selected part count.
1 × 2 = 2
The selected fourth is split the same way.
Write the new name.
1/4 = 2/8
The selected amount is unchanged.
Reverse the split to check.
2 ÷ 2 = 1; 8 ÷ 2 = 4
Pairing restores the original fraction.
Read the proposed chain.
1/2 = 2/4 = 3/8
Every equality requires the same value.
Check the first change.
1 × 2 = 2; 2 × 2 = 4
Both counts doubled correctly.
Inspect the next denominator.
4 × 2 = 8
Fourth-pieces have been halved.
Apply that split to the selected count.
2 × 2 = 4
Both selected fourths become two eighths each.
Repair the final name.
1/2 = 2/4 = 4/8
Four eighths reach the same half-way point.
Check the common number-line position.
All three correct names are halfway from 0 to 1.
Equal position confirms equal value.
Identify the first whole.
Board A: 8 square units
Area defines this whole's size.
Find its selected half.
8 ÷ 2 = 4 square units
Half selects one of two equal areas.
Identify the second whole.
Board B: 4 square units
This whole is smaller.
Find two fourths of the second board.
4 ÷ 4 = 1; 2 × 1 = 2 square units
Two selected fourths each cover one square unit.
Compare the fraction numbers.
1/2 = 2/4
These are equivalent numerical names.
Compare the selected areas.
4 square units > 2 square units
Different whole sizes produce different physical amounts.
State the comparison condition.
Equal fractions give equal amounts when the wholes match.
The missing common whole explains the difference.
Start with six selected eighths.
6/8
Both counts use eighth-pieces.
Pair all pieces of the whole.
8 ÷ 2 = 4
The new pieces are fourths.
Pair the selected pieces.
Write the equivalent name and check.
Complete the pair of equal fractions: 1/6 = ?/18. Type the missing number above the line.
Answer:
Complete the equivalence when 3/4 is repartitioned into twice as many equal parts.
Count every smaller piece in the whole.
4 × 2 = bottom
Each original piece becomes two pieces.
Count the selected smaller pieces.
3 × 2 = top
The selected pieces receive the same split.
Verify the invariant amount.
Pair the smaller pieces to recover the original counts.
Neither the whole nor the selected amount changed.
Which fraction names the same amount as 1/5?
Complete the pair of equal fractions: 3/4 = ?/12. Type the missing number above the line.
Answer:
Which fraction names the same amount as 1/5?
Complete the pair of equal fractions: 1/8 = ?/40. Type the missing number above the line.
Answer:
Two cakes are the same size. Lia takes 1/3 of the first. The second is cut into 9 pieces and Ben takes 3 of them. Express both shares as counts of the smaller pieces, then give the difference between those counts.
First share: first smaller pieces; second share: second smaller pieces; difference: difference pieces
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Complete this chain of equal fraction names: 3/4 = ?/8 = ?/16.
3/4 = eighths/8 = sixteenths/16
You can generate equal fractions and find the simplest form. Without looking: name two fractions equal to 3/4, and say why multiplying both numbers by 5 does not change the amount.
16. Regroup a selected amount, step 3
6 ÷ 2 = 3
Three complete pairs remain selected.
16. Regroup a selected amount, step 4
6/8 = 3/4; 3 × 2 = 6; 4 × 2 = 8
Reversing the regrouping restores both counts.