Grade 3 · 3.NF.A.3 · 3.NF.A.3A

Seeing equivalent fractions in pictures and on the number line

Two thirds of a pan of corn bread is left, and someone cuts every piece in two. Now it is four sixths of the pan. Did the amount of corn bread change, or only its name?

You need

Area and length models of a fractionGrade 3 Locating a fraction on the number lineGrade 3

Unlocks

Adding tenths and hundredths as fractionsGrade 4 Making an equivalent fraction by multiplyingGrade 4

Where you are 4 in Equivalent fractions

Fractions, ratio and percent3

Fractions15

Equivalent fractions4

Seeing equivalent fractions in pictures and on the number line

Making an equivalent fraction by multiplying Common denominators and the least common denominator Simplifying a fraction to simplest form

Why this exists

A fraction names an amount by counting equal pieces: in Area and length models of a fraction, \displaystyle \frac{2}{3} of a pan is 2 of the 3 equal pieces it was cut into. But the pieces are a choice. The same pan could have been cut into 6 pieces, or 9, and the same corn bread would be on it. So one amount can be counted in more than one way, and each way gives it a different name.

Bolor has \displaystyle \frac{2}{3} of a pan of corn bread left. Saran cuts every piece in two, and now the pan holds \displaystyle \frac{4}{6} of its corn bread. Nobody ate any, and nobody baked more. Two different fractions are naming one amount. This lesson is about seeing when that happens, in a picture and on a number line, and making new names for a fraction by cutting every piece in two or by joining pieces in pairs.

The same thing happens on a line. In Locating a fraction on the number line each fraction got a point. A trail marked in fourths of a mile and also in eighths of a mile has a flag at \displaystyle \frac{3}{4} of a mile and a marker at \displaystyle \frac{6}{8} of a mile, and they stand at the same spot. Two names, one point.

Definition

Two fractions are equivalent when they name the same amount of the same whole. On a number line with one unit, that means they name the same point.

We write an equals sign between equivalent fractions:

\frac{2}{3} = \frac{4}{6}

This is read “two thirds equals four sixths”. It does not say the pictures look alike: 3 pieces and 6 pieces are different cuts. It says the amount shaded, or the distance from 0, is the same.

Both parts of the definition need the same whole. Half of a big candy bar and half of a small candy bar are both \displaystyle \frac{1}{2}, and equivalence is a statement about the numbers, measured against one whole. Comparing amounts of two different wholes is a different question (Example 7).

Intuition

The lesson runs over four days. Part 1 cuts every piece of a shaded shape in two, and joins pieces in pairs. Part 2 does it again and again, and finds where joining fails. Part 3 moves to the number line, where equivalent fractions sit on one point. Part 4 keeps the same whole in view and carries the idea past 1.

Part 1: cut every piece in two

Look at the corn bread pan before and after Saran’s cut.

Figure 1The same corn bread before and after every piece is cut in two. The shaded part has not moved.

Saran’s knife runs across the middle of the pan, so it cuts every strip in two. Two things happen at once. The pan had 3 pieces and now has 3 \times 2 = 6. The 2 shaded strips each became 2 pieces, so now 2 \times 2 = 4 pieces are shaded. Both numbers of the fraction doubled.

The shaded corn bread did not change at all; the knife only drew a line through it. That is the whole idea in one picture: the new pieces are half as big, so it takes twice as many of them to cover the same corn bread.

The cut also runs backward. Start with the pan in 6 pieces and 4 shaded. Join the two pieces in each strip back into one, and the pan is in 3 pieces with 2 shaded. Joining pieces in pairs halves both numbers, as long as every shaded piece is joined to another shaded piece.

Part 2: again and again

Nothing stops the cutting after once. A candy bar broken in half is \displaystyle \frac{1}{2} shaded. Cut each half in two and it is in fourths, with \displaystyle \frac{2}{4} shaded. Cut each fourth in two and it is in eighths, with \displaystyle \frac{4}{8} shaded.

Figure 2Cutting every piece in two, and then again. The shaded half of the bar never changes size.

Every row shows the same shaded half, so

\frac{1}{2} = \frac{2}{4} = \frac{4}{8}

Joining is the same walk read upward: 4 eighths pair up into 2 fourths, and 2 fourths pair up into 1 half. But joining only works when the shaded pieces pair up exactly. Three shaded eighths cannot all be joined into fourths: two of them make one fourth, and the third is left beside an unshaded piece. So \displaystyle \frac{3}{8} has no name in fourths.

Part 3: one point on the number line

A number line measures distance from 0, and a fraction is a point on it. Cutting every part of the line in two does to the line what the knife did to the pan.

Figure 3The fourth-mile flags and the eighth-mile markers on one trail. Three fourths and six eighths are the same spot.

The dog sled trail is 1 mile long. Its flags mark every fourth of a mile, and its markers mark every eighth. Each fourth of a mile is split in two by an eighth-mile marker, so the 3 fourths from the start to the third flag are 3 \times 2 = 6 eighths. The dot at \displaystyle \frac{3}{4} and the dot at \displaystyle \frac{6}{8} are one point:

\frac{3}{4} = \frac{6}{8}

This gives a test that needs no picture of shading at all: draw both fractions on lines with the same 0 and the same 1, and see whether they land on the same point.

Part 4: the same whole, and past one

Everything so far kept one whole: one pan, one bar, one mile. That is what makes the names agree. A fourth of a big pan is more corn bread than a fourth of a small pan, so \displaystyle \frac{1}{4} of one and \displaystyle \frac{2}{8} of the other need not be the same amount, even though \displaystyle \frac{1}{4} = \frac{2}{8}.

The number line also runs past 1, and the cutting does not care. On a trail 2 miles long, every fourth of a mile still holds 2 eighths, so \displaystyle \frac{5}{4} of a mile is \displaystyle \frac{10}{8} of a mile. Whole numbers get fraction names this way too: the 1-mile spot is \displaystyle \frac{4}{4} and also \displaystyle \frac{8}{8}. That idea has its own lesson, Whole numbers written as fractions.

Results

Cutting every piece in two. When every piece of a whole is cut into two equal pieces, the number of pieces doubles and the number shaded doubles, and the new fraction is equivalent to the old one. For example \displaystyle \frac{1}{3} = \frac{2}{6} and \displaystyle \frac{3}{4} = \frac{6}{8}.

Joining pairs. When the pieces can be joined in pairs so that every shaded piece is joined to a shaded piece, both numbers halve and the new fraction is equivalent to the old one. For example \displaystyle \frac{2}{8} = \frac{1}{4}.

Same point. Two fractions are equivalent exactly when they name the same point on a number line with the same 0 and the same 1.

Why it is true

Cutting every piece in two. A cut draws a line through the whole without adding or removing any of it, so the shaded amount is what it was. Every old piece is now two new pieces, so the count of all the pieces doubles, and every shaded old piece is two shaded new pieces, so the shaded count doubles. Equal pieces cut in two the same way give equal new pieces, so the new count is still a fraction of equal parts. The amount is the same, so the two fractions are equivalent.

Joining pairs. This is the same cut seen in reverse: if the pairs are joined back, undoing the cut, the shaded amount still does not change.

Same point. On a number line a fraction is the distance from 0, counted in equal jumps. Cutting every jump in two makes twice as many jumps, each half as long, ending at the same place. And two fractions that name the same amount of one unit are the same distance from 0, so they land on one point. Two points that are different are different distances from 0, so they name different amounts. \blacksquare

Worked examples

Examples 1 and 2 go with Part 1, 3 and 4 with Part 2, 5 and 6 with Part 3, and 7 and 8 with Part 4.

Example 1

Part 1: corn bread in thirds, then sixths

Bolor cuts a pan of corn bread into 3 equal strips. His family eats 1 strip, so 2 strips are left. Saran then cuts across the middle of the pan, cutting every strip in two. How many pieces is the pan cut into now, how many are left, and what fraction of the pan is left?

Solution

Before the cut, 2 of the 3 equal strips are left, so \displaystyle \frac{2}{3} of the pan is left. This is the left pan in the figure in Part 1.

Saran’s cut crosses every strip once, so each strip becomes 2 pieces. The pan now has 3 \times 2 = 6 pieces. They are equal, because every strip was equal and each was cut through its middle. Each of the 2 strips that were left also became 2 pieces, so 2 \times 2 = 4 pieces are left. That is \displaystyle \frac{4}{6} of the pan.

No corn bread was eaten or added; the knife only drew lines. So the two fractions name the same amount of the same pan:

\frac{2}{3} = \frac{4}{6}

Check: each sixth is half of a third, so it takes 2 sixths to fill each third, and 4 sixths to fill the 2 thirds that are left.

Example 2

Part 1: joining pieces in pairs

Another day, Bolor’s pan of corn bread is cut into 6 equal pieces in 2 rows of 3. Only 2 pieces are left, one above the other in the left column. Name the part left in sixths, then join the pieces of each column into one and name it again.

Solution

2 of the 6 equal pieces are left, so \displaystyle \frac{2}{6} of the pan is left.

Each column holds 2 pieces, one in each row. Joining the 2 pieces of every column turns the 6 pieces into 6 \div 2 = 3 equal strips, which are thirds. The 2 pieces that are left share a column, so they join into 1 strip, and 1 of the 3 strips is left:

\frac{2}{6} = \frac{1}{3}

The join worked because the 2 pieces left were partners in one column. If the 2 pieces left had been the top piece of the left column and the top piece of the middle column, joining by columns would pair each of them with an eaten piece, and the leftover corn bread would not be a whole number of strips.

Example 3

Part 2: half a candy bar, three ways

Oyun eats half of a candy bar. Show that the amount she ate is also \displaystyle \frac{2}{4} of the bar and \displaystyle \frac{4}{8} of the bar.

Solution

Draw the bar cut into 2 equal pieces with 1 shaded: that is \displaystyle \frac{1}{2}, the top bar in the Part 2 figure.

Cut every piece in two. There are 2 \times 2 = 4 pieces, and the shaded half is now 1 \times 2 = 2 of them. So the half Oyun ate is \displaystyle \frac{2}{4} of the bar.

Cut every piece in two again. There are 4 \times 2 = 8 pieces, and the shaded part is 2 \times 2 = 4 of them, so it is \displaystyle \frac{4}{8} of the bar. The shading never moved, so

\frac{1}{2} = \frac{2}{4} = \frac{4}{8}

Check against the picture: in the bottom bar, 4 of the 8 pieces sit left of the middle and 4 sit right of it, and the left 4 are exactly the shaded half.

Example 4

Part 2: when joining pairs fails

Saran’s candy bar is 8 equal squares in a row. She eats the 2 squares at the left end. Oyun eats 5 squares from the left end of an identical bar. Name each amount in eighths, then name it in fourths if you can.

Solution

Saran ate \displaystyle \frac{2}{8} of her bar and Oyun ate \displaystyle \frac{5}{8} of hers.

To make fourths, join the squares in neighboring pairs: squares 1 and 2, 3 and 4, 5 and 6, 7 and 8. That makes 8 \div 2 = 4 equal pieces.

Saran’s 2 squares are a pair, so they join into 1 piece. She ate 1 of the 4 pieces:

\frac{2}{8} = \frac{1}{4}

Oyun’s 5 squares fill the first two pairs, 2 + 2 = 4 squares, and then square 5 is left over. Its partner, square 6, was not eaten, so it cannot be joined into a shaded fourth. Oyun ate 2 fourths and half of another fourth. No count of fourths names that amount: \displaystyle \frac{2}{4} is too little and \displaystyle \frac{3}{4} is too much. So \displaystyle \frac{5}{8} has no name in fourths.

Example 5

Part 3: the same spot on the trail

The dog sled trail is 1 mile long. Flags stand at every fourth of a mile, and markers stand at every eighth of a mile. Bolor’s team stops at the third flag. Saran’s team stops at the sixth marker. Did they stop at the same spot?

Solution

The third flag is 3 jumps of \displaystyle \frac{1}{4} of a mile from the start, so Bolor stopped at \displaystyle \frac{3}{4} of a mile. The sixth marker is 6 jumps of \displaystyle \frac{1}{8} of a mile, so Saran stopped at \displaystyle \frac{6}{8} of a mile.

There are 4 fourths in the mile and 8 eighths, and 8 = 4 \times 2. The eighth-mile markers cut every fourth of a mile in two, so each jump of \displaystyle \frac{1}{4} is two jumps of \displaystyle \frac{1}{8}. Bolor’s 3 jumps are 3 \times 2 = 6 of the short jumps. The two lines in the Part 3 figure show it: the dot at \displaystyle \frac{3}{4} is straight above the dot at \displaystyle \frac{6}{8}.

They stopped at the same spot, and

\frac{3}{4} = \frac{6}{8}

Check from the far end: the third flag is 4 - 3 = 1 fourth short of the finish, and the sixth marker is 8 - 6 = 2 eighths short. One fourth is two eighths, so both are the same distance from the finish too.

Example 6

Part 3: halves, thirds and sixths

Use the three number lines below (halves, thirds and sixths of one unit) to find every point that has a name on two different lines.

Figure 4Halves, thirds and sixths of the same unit, lined up at 0.

Solution

Thirds and sixths first. There are 3 thirds in the unit and 6 sixths, and 6 = 3 \times 2, so the sixths line is the thirds line with every part cut in two. Every thirds tick is also a sixths tick:

\frac{1}{3} = \frac{2}{6}, \qquad \frac{2}{3} = \frac{4}{6}, \qquad \frac{3}{3} = \frac{6}{6}

Halves and sixths next. Here 6 = 2 \times 3: each half holds 3 sixths, so the halves tick is the third sixths tick, and \displaystyle \frac{1}{2} = \frac{3}{6}. This pair is seen on the line rather than made by cutting in two; each half was cut in three. Also \displaystyle \frac{2}{2} = \frac{6}{6}.

Halves and thirds last. The point \displaystyle \frac{1}{2} is 3 sixths from 0, and the thirds ticks are at 2 sixths and 4 sixths. So \displaystyle \frac{1}{2} falls between \displaystyle \frac{1}{3} and \displaystyle \frac{2}{3}, on no thirds tick. The halves line and the thirds line share only their ends: 0, where \displaystyle \frac{0}{2} = \frac{0}{3}, and 1, where \displaystyle \frac{2}{2} = \frac{3}{3}.

Example 7

Part 4: the same fraction of different pans

Oyun’s corn bread pan is big and Saran’s is small. Oyun eats \displaystyle \frac{1}{4} of her pan. Saran eats \displaystyle \frac{2}{8} of her pan. Are the fractions equivalent? Did they eat the same amount of corn bread?

Solution

The fractions are equivalent. Cut every fourth of any pan in two and it becomes 2 eighths, so \displaystyle \frac{1}{4} = \frac{2}{8}.

But they did not eat the same amount. Saran’s \displaystyle \frac{2}{8} of the small pan is the same amount as \displaystyle \frac{1}{4} of the small pan. A fourth of the big pan is more corn bread than a fourth of the small pan, so Oyun ate more.

Equivalent fractions name the same amount of the same whole. Here the wholes differ, so equal fractions of them are different amounts. Had both eaten from Oyun’s big pan, \displaystyle \frac{1}{4} and \displaystyle \frac{2}{8} of it would be exactly the same corn bread.

Example 8

Part 4: past one mile

A longer dog sled trail is 2 miles long, with flags every fourth of a mile and markers every eighth of a mile. A checkpoint stands at \displaystyle \frac{5}{4} miles. Name the checkpoint in eighths, and name the 1-mile spot and the 2-mile spot on both sets of markings.

Solution

Every fourth of a mile is cut in two by a marker, so each jump of \displaystyle \frac{1}{4} is two jumps of \displaystyle \frac{1}{8}. The checkpoint is 5 fourth-mile jumps from the start, which is 5 \times 2 = 10 eighth-mile jumps:

\frac{5}{4} = \frac{10}{8}

The 1-mile spot is 4 jumps of a fourth, \displaystyle \frac{4}{4}, and 4 \times 2 = 8 jumps of an eighth, \displaystyle \frac{8}{8}. The 2-mile spot is \displaystyle \frac{8}{4} and also \displaystyle \frac{16}{8}, since 8 \times 2 = 16. So \displaystyle 1 = \frac{4}{4} = \frac{8}{8} and \displaystyle 2 = \frac{8}{4} = \frac{16}{8}; these whole-number names are the subject of Whole numbers written as fractions.

Check: the checkpoint is past the 1-mile spot by 5 - 4 = 1 fourth, and by 10 - 8 = 2 eighths. One fourth is two eighths, so both names put it at the same place.

Boundaries

Every piece, the same way. Cutting only some pieces in two leaves pieces of two sizes, and a fraction counts equal pieces. Cut one strip of a 3-strip pan in two and the pan has 4 pieces, but not 4 equal pieces, so it is not in fourths.

Joining needs whole pairs. Pieces can be joined only when every shaded piece is joined to a shaded piece. That is why \displaystyle \frac{2}{8} has a name in fourths and \displaystyle \frac{5}{8} does not.

The same whole. Equivalent fractions name the same amount of one whole, and the same point on one number line. Fractions of different wholes, or points on lines with different units, are not compared this way (Example 7).

Not every pair of cuts shares a point. Halves and thirds of one unit share only 0 and 1. Two fractions whose points differ, such as \displaystyle \frac{1}{2} and \displaystyle \frac{2}{3}, are not equivalent, and no cut makes them so.

Where this goes next

Seeing that one amount wears many names is the start of Equivalent fractions, and it widens Whole numbers written as fractions, where 1 is \displaystyle \frac{4}{4} and also \displaystyle \frac{8}{8}. Points on one number line also make comparing possible: of two fractions on the same line, the one farther right is greater, and when the pieces are the same size the fraction with more of them is greater, as in Comparing fractions with the same denominator or numerator. Later, cutting every piece into 3, 4 or more equal pieces instead of 2 becomes Making an equivalent fraction by multiplying, and joining pieces in groups becomes Simplifying a fraction to simplest form. Two fractions with different pieces are then renamed in one shared size of piece, which is Common denominators and the least common denominator, the key to adding fractions like \displaystyle \frac{1}{2} and \displaystyle \frac{1}{4}.

Exercises

Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 17 with Part 3, and 18 to 21 with Part 4. The reasoning group draws on all four.

Part 1: cut every piece in two

A pan of corn bread is cut into 2 equal pieces, and 1 piece is left. Saran cuts every piece in two. How many pieces does the pan have now, how many are left, and what fraction is that?
A pan of corn bread is cut into 4 equal pieces, and 3 are left. Bolor cuts every piece in two. Name the part left before and after his cut.
A pan of corn bread is cut into 8 equal strips side by side. The 6 strips at the left end are left. Join neighboring strips in pairs and name the part left. Then try joining the new pieces in pairs again: can the part left be named in halves?
In picture A, name the shaded part. Then imagine one cut across the middle of the rectangle and name the shaded part again.
In picture B, name the shaded part. Then join the two pieces of every column into one and name it again.
In picture D, name the shaded part. Then join the two pieces of every column into one and name it again.

Figure 5Pictures for Exercises 4 to 7.

Part 2: again and again

In picture C, name the shaded part. Can it be named in fourths by joining the two pieces of each column? Explain.
Fill in each box, and draw a candy bar picture that shows it. (a) \displaystyle \frac{1}{4} = \frac{\square}{8} (b) \displaystyle \frac{1}{3} = \frac{\square}{6} (c) \displaystyle \frac{\square}{4} = \frac{6}{8} (d) \displaystyle \frac{4}{\square} = \frac{2}{3}
Bolor breaks a candy bar into 4 equal pieces and eats 1. Oyun breaks an identical candy bar into 8 equal pieces and eats 2. Did they eat the same amount? Explain with cutting in two.
Which of \displaystyle \frac{2}{8}, \displaystyle \frac{4}{8}, \displaystyle \frac{6}{8} and \displaystyle \frac{7}{8} can be named in fourths? Give the fourths name for each one that can.
Which of these equal \displaystyle \frac{1}{2}: \displaystyle \frac{2}{4}, \displaystyle \frac{3}{6}, \displaystyle \frac{3}{8}, \displaystyle \frac{4}{8}, \displaystyle \frac{2}{6}?
A candy bar is cut into 6 equal pieces. Saran eats 2 pieces and Bolor eats 2 pieces. Name each one’s share and the part left, in sixths and then in thirds.

Part 3: one point on the number line

Draw a number line from 0 to 1 in halves and, under it, one of the same length in fourths. List every point that has a name on both lines.
On the 1-mile dog sled trail with flags every fourth of a mile and markers every eighth, Oyun’s team stops at the first flag. At which marker is that, and what is the equivalent fraction?
A number line from 0 to 1 is cut into fourths, and a dot sits on the second tick after 0. Name the point in fourths, in halves and in eighths.
On the same trail, Saran’s team stops at \displaystyle \frac{5}{8} of a mile and Bolor’s team at \displaystyle \frac{3}{4} of a mile. Did they stop at the same spot? If not, who went farther, and by how much?
Which fractions with 8 on the bottom, from \displaystyle \frac{0}{8} to \displaystyle \frac{8}{8}, sit on the same point as a tick of a fourths line?

Part 4: the same whole, and past one

On the 2-mile trail with flags every fourth of a mile and markers every eighth, a water stop stands at \displaystyle \frac{7}{4} miles. Name it in eighths.
On a number line from 0 to 2, fill in the boxes: \displaystyle \frac{3}{2} = \frac{\square}{4} = \frac{\square}{8}.
Bolor eats \displaystyle \frac{1}{2} of a big candy bar. Saran eats \displaystyle \frac{2}{4} of a small candy bar. Are the fractions equivalent? Did they eat the same amount of candy?
A number line from 0 to 2 is marked in thirds and also in sixths. Name the point 1 and the point 2 in thirds and in sixths.

Reasoning

Explain why cutting every piece of a shaded shape in two doubles the number of pieces and the number shaded, but does not change the amount shaded.
Is it always possible to join the pieces of a shaded picture in pairs? Give an example where it works and one where it does not, and say what makes the difference.
Explain why two fractions on the same number line that land on the same point must name the same amount of the unit.
A pan of corn bread is cut into 3 equal strips, and then only 1 strip is cut in two. The pan now has 4 pieces. Is it cut into fourths? Explain.

Answers

4 pieces, 2 left, \displaystyle \frac{2}{4}. The cut makes 2 \times 2 = 4 pieces, and the 1 piece left becomes 1 \times 2 = 2, so \displaystyle \frac{1}{2} = \frac{2}{4}.
Before: \displaystyle \frac{3}{4}. After: 4 \times 2 = 8 pieces with 3 \times 2 = 6 left, so \displaystyle \frac{3}{4} = \frac{6}{8}.
\displaystyle \frac{6}{8} to start. Joining pairs gives 8 \div 2 = 4 pieces with 6 \div 2 = 3 left, so \displaystyle \frac{6}{8} = \frac{3}{4}. Joining again fails: the 3 fourths left make one pair, a half, and the third fourth’s partner was eaten. So \displaystyle \frac{3}{4} has no name in halves.
\displaystyle \frac{1}{3}. The cut makes 3 \times 2 = 6 pieces with 1 \times 2 = 2 shaded, so \displaystyle \frac{1}{3} = \frac{2}{6}.
\displaystyle \frac{6}{8}. Joining columns gives 8 \div 2 = 4 pieces; the 3 shaded columns are whole, so 3 are shaded: \displaystyle \frac{6}{8} = \frac{3}{4}.
\displaystyle \frac{2}{4}. Joining columns gives 2 pieces with the left one shaded: \displaystyle \frac{2}{4} = \frac{1}{2}.
\displaystyle \frac{3}{8}. No: each shaded piece’s column partner, the piece below it, is not shaded, so no column is wholly shaded. It also has no fourths name at all, since joining neighbors in a row leaves the third shaded piece beside an unshaded one.
(a) 2, since cutting each fourth in two makes 4 \times 2 = 8 pieces and 1 \times 2 = 2 shaded. (b) 2, since 3 \times 2 = 6 and 1 \times 2 = 2. (c) 3, since joining pairs of 8 eighths gives 4 pieces and 6 \div 2 = 3 shaded. (d) 6, since cutting each third in two gives 3 \times 2 = 6 pieces.
Yes. Cutting each of Bolor’s 4 pieces in two makes 8 pieces, and his 1 piece becomes 2, so \displaystyle \frac{1}{4} = \frac{2}{8}. The bars are identical, so it is the same amount of candy.
\displaystyle \frac{2}{8} = \frac{1}{4}, \displaystyle \frac{4}{8} = \frac{2}{4} and \displaystyle \frac{6}{8} = \frac{3}{4}, since 2, 4 and 6 squares from the end join into 1, 2 and 3 pairs. \displaystyle \frac{7}{8} cannot: 7 squares make 3 pairs and one square left over.
\displaystyle \frac{2}{4}, \displaystyle \frac{3}{6} and \displaystyle \frac{4}{8}. Each is half of its pieces: 2 + 2 = 4, 3 + 3 = 6, 4 + 4 = 8. \displaystyle \frac{3}{8} is less than half (4 would be half) and so is \displaystyle \frac{2}{6} (3 would be half).
Saran \displaystyle \frac{2}{6} = \frac{1}{3}, Bolor \displaystyle \frac{2}{6} = \frac{1}{3}, and 6 - 2 - 2 = 2 pieces left, \displaystyle \frac{2}{6} = \frac{1}{3}. Joining pairs makes 6 \div 2 = 3 pieces, each person and the part left getting 1, if each one’s 2 pieces sit side by side.
0, then \displaystyle \frac{1}{2} = \frac{2}{4}, then \displaystyle \frac{2}{2} = \frac{4}{4}, which is 1. The fourths line is the halves line with each half cut in two.
The second marker: \displaystyle \frac{1}{4} = \frac{2}{8}, since 1 \times 2 = 2.
\displaystyle \frac{2}{4}, which is \displaystyle \frac{1}{2} (join pairs: 4 \div 2 = 2 parts, 1 reached) and \displaystyle \frac{4}{8} (cut in two: 2 \times 2 = 4 jumps of an eighth).
No. \displaystyle \frac{3}{4} = \frac{6}{8}, and 6 - 5 = 1, so Bolor went \displaystyle \frac{1}{8} of a mile farther.
\displaystyle \frac{0}{8}, \displaystyle \frac{2}{8}, \displaystyle \frac{4}{8}, \displaystyle \frac{6}{8} and \displaystyle \frac{8}{8}: every second eighth, since each fourth holds 2 eighths.
\displaystyle \frac{14}{8} miles, since 7 \times 2 = 14.
\displaystyle \frac{3}{2} = \frac{6}{4} = \frac{12}{8}: each half is 2 fourths, 3 \times 2 = 6, and each fourth is 2 eighths, 6 \times 2 = 12.
The fractions are equivalent, \displaystyle \frac{1}{2} = \frac{2}{4}, but they did not eat the same amount. The wholes differ: half of the big bar is more candy, so Bolor ate more.
\displaystyle 1 = \frac{3}{3} = \frac{6}{6} and \displaystyle 2 = \frac{6}{3} = \frac{12}{6}, since 3 \times 2 = 6 and 6 \times 2 = 12.
The cuts only draw lines through the shape, so no shaded part is added or taken away. Every piece turns into 2 pieces, so the count of all pieces doubles; every shaded piece turns into 2 shaded pieces, so the shaded count doubles. The new pieces are half as big, so twice as many cover the same amount.
No. It works when every shaded piece can be joined to a shaded partner, as with \displaystyle \frac{4}{6} in 2 rows of 3 with two whole columns shaded, which joins to \displaystyle \frac{2}{3}. It fails when a shaded piece’s partner is unshaded, as with \displaystyle \frac{3}{8} of a bar of 8 squares, which has one shaded square left over.
A point on the line is a distance from 0. Two fractions on the same point are the same distance from 0, measured with the same unit, so they are the same amount of that unit.
No. The pan has 4 pieces, but 2 are half strips and 2 are whole strips, so they are not equal. Fourths are 4 equal pieces. To keep the fraction names working, every strip must be cut in two, which makes sixths.