{"name":"Slices","url":"https://slices.org/","description":"Free three-minute lessons on the 20 fraction ideas that trip people up, for kids 8 to 12, the grown-ups helping them, and anyone who never quite got fractions.","audience":"Children aged 8–12 (grades 3–6), parents, teachers and adults","access":"Free. No sign-up required. No ads.","llms":"https://slices.org/llms.txt","families":[{"id":"parts-and-wholes","number":1,"title":"Parts & Wholes","aim":"What fractions describe.","description":"Equal shares of a whole, and what each number in a fraction is counting. Start here if fractions feel shaky: every other idea rests on these.","url":"https://slices.org/explore/parts-and-wholes"},{"id":"fraction-size","number":2,"title":"Fractions Have Size","aim":"Fractions are numbers with a size.","description":"Why 1/8 is smaller than 1/6, why 3/8 is less than 2/3, and how fractions live on the number line, including fractions greater than one.","url":"https://slices.org/explore/fraction-size"},{"id":"equivalence-and-comparison","number":3,"title":"Same Value, Different Forms","aim":"Equivalence and comparison.","description":"How 1/2 and 2/4 can be the same number, why simplifying never shrinks a fraction, and how to compare fractions by reasoning instead of guessing.","url":"https://slices.org/explore/equivalence-and-comparison"},{"id":"working-with-fractions","number":4,"title":"Working With Fractions","aim":"What fraction operations mean.","description":"Why we need common units to add and subtract, why multiplying can make a number smaller, and why dividing can make it bigger.","url":"https://slices.org/explore/working-with-fractions"}],"slices":[{"number":1,"id":"s01","slug":"equal-means-equal","title":"Equal Means Equal","family":"parts-and-wholes","question":"Do the parts of a fraction have to be equal?","answer":"Yes. A fraction like 1/4 means one of 4 equal parts of a whole, so 4 pieces of different sizes are not fourths. The parts must be the same size, but they don’t have to be the same shape.","confusion":"Any four pieces are fourths, even when the pieces are different sizes.","overheard":"I cut it into four pieces, so each piece is a fourth.","insight":"Fractions need equal shares of one whole. Four parts are fourths only when all four are the same size, even if their shapes differ.","summary":"Why four pieces are not always fourths: a fraction needs equal shares of the whole.","minutes":3,"standards":["3.G.A.2","2.G.A.3","3.NF.A.1"],"url":"https://slices.org/slices/equal-means-equal","markdown":"https://slices.org/slices/equal-means-equal.md"},{"number":2,"id":"s02","slug":"whats-the-whole","title":"What’s the Whole?","family":"parts-and-wholes","question":"Is half of a large pizza the same amount as half of a small pizza?","answer":"No. Half of a large pizza is more pizza than half of a small pizza. A fraction describes part of a particular whole, so the same fraction of a bigger whole is a bigger amount.","confusion":"One half is always the same amount, so half of a small pizza is as much as half of a large one.","overheard":"Half is half. My half of the little pizza is the same as your half of the big one.","insight":"A fraction describes part of a particular whole. The same fraction of different wholes can be a different amount.","summary":"Why half of a big pizza is more than half of a small one: a fraction depends on its whole.","minutes":3,"standards":["3.NF.A.3.d","3.NF.A.1","4.NF.A.2"],"url":"https://slices.org/slices/whats-the-whole","markdown":"https://slices.org/slices/whats-the-whole.md"},{"number":3,"id":"s03","slug":"the-bottom-number","title":"The Bottom Number","family":"parts-and-wholes","question":"What does the bottom number of a fraction mean?","answer":"The bottom number, the denominator, tells how many equal parts make one whole. In 1/5, the whole is cut into 5 equal parts. It counts every part, shaded or not, so one shaded part out of five is 1/5.","confusion":"The bottom number counts the shaded pieces, or the pieces left over, so one shaded piece out of five gets written 5/1 or 1/4.","overheard":"One piece is shaded and four aren’t, so that’s 1/4.","insight":"The bottom number tells how many equal parts make one whole. It counts every part, shaded or not.","summary":"What the bottom number of a fraction counts: all the equal parts that make one whole.","minutes":3,"standards":["3.NF.A.1","3.NF.A.2.a"],"url":"https://slices.org/slices/the-bottom-number","markdown":"https://slices.org/slices/the-bottom-number.md"},{"number":4,"id":"s04","slug":"the-top-number","title":"The Top Number","family":"parts-and-wholes","question":"What does the top number of a fraction mean?","answer":"The top number, the numerator, counts how many of the equal parts you have. In 3/5, the whole is cut into 5 equal parts and the 3 says you have three of them.","confusion":"The top number counts all the parts, or the parts left over, instead of the parts we have.","overheard":"The top number is how many pieces there are altogether.","insight":"The top number counts how many of the equal parts we have. In 3/5, we have 3 of the 5 equal parts.","summary":"What the top number of a fraction counts: how many of the equal parts you have.","minutes":3,"standards":["3.NF.A.1","4.NF.B.3","3.NF.A.2.b"],"url":"https://slices.org/slices/the-top-number","markdown":"https://slices.org/slices/the-top-number.md"},{"number":5,"id":"s05","slug":"fractions-of-collections","title":"Fractions of Collections","family":"parts-and-wholes","question":"What is 1/4 of 12?","answer":"1/4 of 12 is 3. Share the 12 into 4 equal groups and each group has 3, so one group is one fourth of the whole collection.","confusion":"Fractions only describe pieces of one object, so 1/4 of 12 counters must be 1 counter, or 4 counters.","overheard":"1/4 of 12 counters is 1 counter. It says 1.","insight":"A whole can be a group of things. 1/4 of 12 is one of 4 equal groups, which is 3.","summary":"How to find a fraction of a group, and why 1/4 of 12 counters is 3, not 1 or 4.","minutes":3,"standards":["3.NF.A.1","5.NF.B.4.a"],"url":"https://slices.org/slices/fractions-of-collections","markdown":"https://slices.org/slices/fractions-of-collections.md"},{"number":6,"id":"s06","slug":"bigger-denominator-trap","title":"The Bigger Denominator Trap","family":"fraction-size","question":"Is 1/6 bigger than 1/8?","answer":"Yes. 1/6 is bigger than 1/8. When the same whole is cut into 8 equal pieces instead of 6, each piece has to be smaller, so one eighth is less than one sixth.","confusion":"A larger denominator means a larger piece, so 1/8 must be bigger than 1/6.","overheard":"Eight is more than six, so 1/8 is bigger.","insight":"When the whole stays the same, more equal pieces means smaller pieces.","summary":"Why 1/8 is smaller than 1/6, even though 8 is bigger than 6.","minutes":3,"standards":["3.NF.A.3.d"],"url":"https://slices.org/slices/bigger-denominator-trap","markdown":"https://slices.org/slices/bigger-denominator-trap.md"},{"number":7,"id":"s07","slug":"bigger-numerator-trap","title":"The Bigger Numerator Trap","family":"fraction-size","question":"Which is bigger, 3/8 or 2/3?","answer":"2/3 is bigger. 3/8 is three small parts (eighths), which is less than half of the whole. 2/3 is two much bigger parts (thirds), which is more than half. So 2/3 > 3/8, even though 3 is bigger than 2.","confusion":"A larger numerator means a larger fraction, so 3/8 must be bigger than 2/3.","overheard":"3/8 has a 3 and 2/3 only has a 2, so 3/8 is bigger.","insight":"A fraction’s size depends on both numbers together: the top counts the parts, and the bottom sets how big each part is.","summary":"Why 2/3 is bigger than 3/8, even though 3 is bigger than 2.","minutes":3,"standards":["4.NF.A.2"],"url":"https://slices.org/slices/bigger-numerator-trap","markdown":"https://slices.org/slices/bigger-numerator-trap.md"},{"number":8,"id":"s08","slug":"more-than-one-whole","title":"More Than One Whole","family":"fraction-size","question":"Can a fraction be bigger than 1?","answer":"Yes. A fraction is bigger than 1 when its top number is bigger than its bottom number. 6/4 means six fourths: four fourths make one whole, and two more fourths make half of another, so 6/4 = 1 1/2.","confusion":"Fractions are always less than one, so a fraction such as 6/4 cannot be a real number.","overheard":"6/4 can’t be real. There are only 4 fourths in a pizza.","insight":"A fraction can be less than, equal to or more than one whole. Four fourths make one whole, so six fourths is one whole and two fourths more.","summary":"Why 6/4 makes sense: a fraction can equal one whole or go beyond it.","minutes":3,"standards":["3.NF.A.3.c","4.NF.B.3","3.NF.A.2.b"],"url":"https://slices.org/slices/more-than-one-whole","markdown":"https://slices.org/slices/more-than-one-whole.md"},{"number":9,"id":"s09","slug":"counting-spaces-not-ticks","title":"Counting Spaces, Not Ticks","family":"fraction-size","question":"How do you find 3/4 on a number line?","answer":"Split the line from 0 to 1 into 4 equal spaces, then count 3 spaces from 0. The tick at the end of the third space is 3/4. Count spaces, not tick marks: 4 equal spaces need 5 tick marks.","confusion":"A fraction’s place on a number line is found by counting tick marks, so the tick at 0 gets counted as “one”, or the number of ticks is taken as the denominator.","overheard":"There are 5 marks between 0 and 1, so these must be fifths.","insight":"The denominator tells how many equal spaces split 0 to 1. The numerator counts the spaces you hop from 0, not the tick marks.","summary":"Why we count equal spaces from 0, not tick marks, to find 3/4 on a number line.","minutes":3,"standards":["3.NF.A.2.a","3.NF.A.2.b"],"url":"https://slices.org/slices/counting-spaces-not-ticks","markdown":"https://slices.org/slices/counting-spaces-not-ticks.md"},{"number":10,"id":"s10","slug":"one-number-not-two","title":"One Number, Not Two","family":"fraction-size","question":"Where is 2/5 on a number line?","answer":"2/5 is between 0 and 1, a little less than 1/2. Split the space from 0 to 1 into 5 equal parts and count 2 parts from 0. A fraction is one number with one place on the line, not a 2 and a 5.","confusion":"A fraction is two separate whole numbers, so 2/5 belongs somewhere near 2 or 5 on the number line.","overheard":"2/5 goes between the 2 and the 5 on the number line.","insight":"A fraction is one number with one place on the number line. 2/5 is two fifths of the way from 0 to 1.","summary":"Why 2/5 is one number between 0 and 1, not a 2 and a 5.","minutes":3,"standards":["3.NF.A.2","3.NF.A.2.b"],"url":"https://slices.org/slices/one-number-not-two","markdown":"https://slices.org/slices/one-number-not-two.md"},{"number":11,"id":"s11","slug":"different-numbers-same-amount","title":"Different Numbers, Same Amount","family":"equivalence-and-comparison","question":"Is 1/2 the same as 2/4?","answer":"Yes. 1/2 and 2/4 are the same amount. Split each half into 2 equal pieces and the whole has 4 equal pieces, with 2 of them shaded. The pieces are half the size and there are twice as many, so the amount does not change. Both fractions sit at the same point on the number line.","confusion":"Fractions written with different numbers must be different amounts, so 1/2 and 2/4 can’t be equal. Often 2/4 seems bigger, because its numbers are bigger.","overheard":"2/4 is bigger than 1/2. The numbers are bigger.","insight":"Different numbers can name the same amount. Split every piece into the same number of equal pieces and the amount stays the same: 1/2 = 2/4 = 4/8.","summary":"Why 1/2, 2/4 and 4/8 are exactly the same amount, even though the numbers are different.","minutes":3,"standards":["3.NF.A.3.a","3.NF.A.3.b","4.NF.A.1"],"url":"https://slices.org/slices/different-numbers-same-amount","markdown":"https://slices.org/slices/different-numbers-same-amount.md"},{"number":12,"id":"s12","slug":"simplifying-doesnt-shrink","title":"Simplifying Doesn’t Shrink","family":"equivalence-and-comparison","question":"Does simplifying a fraction make it smaller?","answer":"No. Simplifying changes how a fraction is written, not how big it is. To simplify 6/8, group the eighths in pairs: each pair is one fourth, so 6/8 becomes 3/4. The pieces are bigger and there are fewer of them, but the amount is exactly the same.","confusion":"Simplifying makes a fraction smaller: 3/4 must be less than 6/8, because 3 and 4 are smaller numbers than 6 and 8.","overheard":"If you simplify 6/8 to 3/4, you lose some.","insight":"Simplifying changes the pieces you count with, not the amount. Group six eighths in pairs and you get three fourths: bigger pieces, fewer of them, the same amount.","summary":"Why 6/8 and 3/4 are the same amount, and why simplifying a fraction never makes it smaller.","minutes":3,"standards":["4.NF.A.1","3.NF.A.3.b"],"url":"https://slices.org/slices/simplifying-doesnt-shrink","markdown":"https://slices.org/slices/simplifying-doesnt-shrink.md"},{"number":13,"id":"s13","slug":"same-bottom-compare-the-top","title":"Same Bottom, Compare the Top","family":"equivalence-and-comparison","question":"How do you compare fractions with the same denominator?","answer":"Compare the numerators. When the denominators match, both fractions are made of the same-size pieces, so the one with more pieces is bigger. For example, 5/7 > 2/7, because five sevenths is more than two sevenths of the same whole.","confusion":"Fractions with the same bottom number must be the same size, because both are “sevenths”. Or the smaller top number makes the bigger fraction, the way the smaller bottom number did for 1/6 and 1/8.","overheard":"2/7 and 5/7 are both sevenths, so they’re the same size.","insight":"When the bottom numbers match, the pieces are the same size, so count them. More of the same pieces is more: 5/7 > 2/7.","summary":"Why 5/7 is bigger than 2/7: the pieces are the same size, so the one with more pieces is bigger.","minutes":3,"standards":["3.NF.A.3.d"],"url":"https://slices.org/slices/same-bottom-compare-the-top","markdown":"https://slices.org/slices/same-bottom-compare-the-top.md"},{"number":14,"id":"s14","slug":"same-top-compare-the-pieces","title":"Same Top, Compare the Pieces","family":"equivalence-and-comparison","question":"How do you compare fractions with the same numerator?","answer":"Compare the size of the pieces. When the numerators match, both fractions have the same number of pieces, so the one with bigger pieces (the smaller denominator) is bigger. For example, 3/5 > 3/8, because fifths are bigger than eighths.","confusion":"With the same top number, the fraction with the bigger bottom number is bigger (3/8 > 3/5, because 8 > 5), or the two are equal because both have 3 pieces.","overheard":"3/5 and 3/8 both have 3 pieces, so they’re equal.","insight":"When the top numbers match, both fractions have the same number of pieces, so compare the pieces. Fewer parts in the whole means bigger pieces: 3/5 > 3/8.","summary":"Why 3/5 is bigger than 3/8: the same number of pieces, but fifths are bigger pieces than eighths.","minutes":3,"standards":["3.NF.A.3.d"],"url":"https://slices.org/slices/same-top-compare-the-pieces","markdown":"https://slices.org/slices/same-top-compare-the-pieces.md"},{"number":15,"id":"s15","slug":"compare-without-guessing","title":"Compare Without Guessing","family":"equivalence-and-comparison","question":"Which is bigger, 3/8 or 4/7?","answer":"4/7 is bigger. Half of 8 is 4, so 3/8 is less than one half. Half of 7 is three and a half, so 4/7 is more than one half. A fraction above one half is always bigger than a fraction below it.","confusion":"When the tops and bottoms are both different, fractions can only be compared by guessing or by the digits: 3/8 must be bigger than 4/7, because 8 is bigger than 7.","overheard":"The tops and bottoms are all different, so I just guessed.","insight":"Compare by position, not digits. 3/8 is less than one half and 4/7 is more than one half, so 4/7 is bigger.","summary":"How to compare fractions like 3/8 and 4/7 by asking which side of one half each one is on.","minutes":4,"standards":["4.NF.A.2"],"url":"https://slices.org/slices/compare-without-guessing","markdown":"https://slices.org/slices/compare-without-guessing.md"},{"number":16,"id":"s16","slug":"adding-different-pieces","title":"Adding Different Pieces","family":"working-with-fractions","question":"Is 1/2 + 1/4 equal to 2/6?","answer":"No. 1/2 + 1/4 = 3/4. One half is the same as two fourths, so one half plus one fourth makes three fourths. 2/6 is less than 1/2, so it can’t be the total.","confusion":"To add fractions, add the top numbers and add the bottom numbers, so 1/2 + 1/4 = 2/6.","overheard":"1/2 + 1/4 is 2/6. You add the tops and add the bottoms.","insight":"Pieces can only be counted together when they are the same size. Change to a common unit first: 1/2 + 1/4 = 2/4 + 1/4 = 3/4.","summary":"Why 1/2 + 1/4 is 3/4, not 2/6: make the pieces the same size before you count them.","minutes":3,"standards":["5.NF.A.1","5.NF.A.2"],"url":"https://slices.org/slices/adding-different-pieces","markdown":"https://slices.org/slices/adding-different-pieces.md"},{"number":17,"id":"s17","slug":"keep-the-unit-when-adding","title":"Keep the Unit When Adding","family":"working-with-fractions","question":"Is 1/5 + 2/5 equal to 3/10?","answer":"No. 1/5 + 2/5 = 3/5. Joining one fifth and two fifths gives three fifths: the number of pieces changes, but each piece is still a fifth. 3/10 is only half as much as 3/5.","confusion":"When fractions have the same denominator, add the denominators too, so 1/5 + 2/5 = 3/10.","overheard":"1/5 + 2/5 is 3/10, because five plus five is ten.","insight":"Adding like parts changes how many parts there are, not their size: 1/5 + 2/5 = 3/5.","summary":"Why 1/5 + 2/5 is 3/5, not 3/10: joining fifths gives more fifths, not smaller pieces.","minutes":3,"standards":["4.NF.B.3","4.NF.B.3.a","4.NF.B.3.d"],"url":"https://slices.org/slices/keep-the-unit-when-adding","markdown":"https://slices.org/slices/keep-the-unit-when-adding.md"},{"number":18,"id":"s18","slug":"taking-pieces-away","title":"Taking Pieces Away","family":"working-with-fractions","question":"Is 3/4 − 1/2 equal to 2/2?","answer":"No. 3/4 − 1/2 = 1/4. One half is the same as two fourths, so taking it away from three fourths leaves one fourth. 2/2 is a whole, which is more than the 3/4 we started with.","confusion":"To subtract fractions, subtract the top numbers and the bottom numbers, so 3/4 − 1/2 = 2/2.","overheard":"3/4 take away 1/2 is 2/2. Three minus one, four minus two.","insight":"Subtraction takes away an amount. Use the same-size pieces for both amounts, then take away: 3/4 − 1/2 = 3/4 − 2/4 = 1/4.","summary":"Why 3/4 − 1/2 is 1/4, not 2/2: take away an amount using same-size pieces.","minutes":3,"standards":["5.NF.A.1","5.NF.A.2","4.NF.B.3.a"],"url":"https://slices.org/slices/taking-pieces-away","markdown":"https://slices.org/slices/taking-pieces-away.md"},{"number":19,"id":"s19","slug":"multiplication-can-make-smaller","title":"Multiplication Can Make Smaller","family":"working-with-fractions","question":"Why does multiplying by a fraction make a number smaller?","answer":"Because multiplying by a fraction less than one takes only part of an amount. 1/2 × 1/2 means one half of one half, which is 1/4: smaller than the 1/2 we started with.","confusion":"Multiplying always makes a number bigger, so 1/2 × 1/2 can’t be smaller than 1/2.","overheard":"Times always makes it bigger, so 1/2 × 1/2 has to be more than 1/2.","insight":"Multiplying by a fraction less than one takes only part of an amount, so the answer is smaller: 1/2 × 1/2 = 1/4.","summary":"Why 1/2 × 1/2 is 1/4: multiplying by a fraction less than one takes part of an amount.","minutes":4,"standards":["5.NF.B.4.a","5.NF.B.4.b","5.NF.B.5.b"],"url":"https://slices.org/slices/multiplication-can-make-smaller","markdown":"https://slices.org/slices/multiplication-can-make-smaller.md"},{"number":20,"id":"s20","slug":"division-can-make-bigger","title":"Division Can Make Bigger","family":"working-with-fractions","question":"Why does dividing by a fraction give a bigger answer?","answer":"Because dividing by a fraction asks how many of those pieces fit into an amount. Pieces smaller than one whole fit more than once into every whole, so the count is bigger than the amount: two fourths fit into one half, so 1/2 ÷ 1/4 = 2.","confusion":"Dividing always makes a number smaller, so 1/2 ÷ 1/4 must be less than 1/2.","overheard":"Dividing makes things smaller, so 1/2 ÷ 1/4 is less than 1/2.","insight":"Dividing by a fraction asks how many of those pieces fit. When the pieces are smaller than one whole, the answer is bigger than the amount: 1/2 ÷ 1/4 = 2.","summary":"Why 1/2 ÷ 1/4 is 2: dividing by a fraction asks how many of those pieces fit.","minutes":3,"standards":["6.NS.A.1","5.NF.B.7.b"],"url":"https://slices.org/slices/division-can-make-bigger","markdown":"https://slices.org/slices/division-can-make-bigger.md"}]}