#1: 6 June 2022

The gray area is what fraction of the white area?

This task is inspired by Pat Thompson's "Three Fifths Problem, which appears on p. 103 of NCTM's Making Sense of Fractions, Ratios, and Proportions.

#2: 7 June 2022

The four smaller squares are equal in area to the larger one. If 3/4 of the entire figure gets shaded, how many regions will be shaded? (assuming no new lines are drawn)

#3: 8 June 2022

Which colored region corresponds to the sum, 1/6 + 1/3?

#4: 9 June 2022

2/3 is what fraction (i.e., what part) of 2?

#5: 10 June 2022

If each kid gets 3/4 of one waffle, how many waffles are needed to feed 4 kids?

#6: 11 June 2022

What you see here is 2/3 of an order of blobs. How many blobs are in 1/2 an order?

#7: 12 June 2022

The shaded region is what fraction of the entire rectangle?

#8: 13 June 2022

Which letter corresponds to the location of 9/8 on this number line?

This task is adapted from one by Susan Lamon on p. 222 of Teaching Fractions and Ratios for Understanding.

#9: 14 June 2022

Do you see 2/3 (“two thirds”) of this triangle? How much of the triangle is 3/2 (“three halves”) of that 2/3?

#10: 15 June 2022

These 8 daisies represent 1/3 more than a bunch of daisies. How many daisies are in 1 bunch?

#11: 16 June 2022

These are the 2 cakes that were ordered for Kim’s birthday. First, Kim took 1/6 of the 2 cakes. Then Paolo took 1/5 of what remained.

• Who took the bigger share, Kim or Paolo?

• Why can’t we just compare 1/6 and 1/5 to find the answer?

#12: 17 June 2022

## How long is this leaf?

#13: 18 June 2022

A multiplication or division operation was applied to the first set of waffles to produce the number of waffles in the second set. Then a second multiplication or division operation was applied to the second set to produce the number of waffles in the third set. There is a fraction that "contains" these two operations that can be multiplied by the number of waffles in the first set to produce the number in the third set. What is it?

#14: 19 June 2022

If there are 16 oranges, how many apples and bananas are there?

This task is adapted from one by Susan Lamon on p. 206 of Teaching Fractions and Ratios for Understanding.

#15: 20 June 2022

Assume that altogether, the glasses in Set A hold the same volume of liquid as the glasses in Set B. A small glass contains what fraction of the volume of a large glass?

I found my answer by removing the same glasses from each set. Then I looked at what was left over.

#16: 21 June 2022

What values of A, B, and C make it so that this figure can be used to represent the equation, A/4 + B/36 + C/9 = 1 ?

#17: 22 June 2022

What sequence of folds of a standard sheet of paper will divide the paper into equal-size regions, each of which is 1/24th of the size of the original sheet of paper?

I started by folding the paper in 1/2 and then in 1/2 again. That divided the paper into 2 × 2 = 4 regions.

#18: 23 June 2022

This one's a classic ...

A brick weighs one pound plus the weight of half the brick. How many pounds does the brick weigh?

My answer isn't 1.5 pounds. :)

#19: 24 June 2022

This division expression can be interpreted as, “How many 5/6s are there in 2 and 1/2?” Use the hexagons to find the answer.

#20: 25 June 2022

Do you see a region you could call 3/5 (“3 fifths”)? What region corresponds to 5/3 (“5 thirds”) of that 3/5?

This task comes from Pat Thompson's "Three Fifths Problem, which appears on p. 103 of NCTM's Making Sense of Fractions, Ratios, and Proportions.

In order to find 5/3 (“5 thirds”) of 3/5, I first found just 1/3 (“1 third”) of 3/5.

#21: 26 June 2022

How many times will 1/3 of this candy bar fit into 1/2 of this candy bar?

#22: 27 June 2022

3/5 of a book collection fills up 2/3 of a bookshelf. How much of the collection will fill the entire bookshelf?

It helped me to start by figuring out how much of the collection fills up 1/3 of the bookshelf.

#23: 28 June 2022

What positive numbers can fill the three boxes so that the three fractions add up to 1?

There are many correct answers to this problem.

I found three fractions that would cover the entire width of the rectangle from left to right.

#24: 29 June 2022

• How far is each fraction (9/10 and 10/9) from 1 ? Give your answers in the form of a unit fraction. A unit fraction has a numerator of 1, like 1/2 and 1/5.

• Which fraction is closer to 1?

I began by noting that 9/9 and 10/10 are equal to 1.

#25: 30 June 2022

Use each of the numbers 1, 2, 3, and 4 only once to construct the smallest positive difference between two fractions. Then do the same thing for the largest positive difference.

I found it useful to think of the difference between two numbers as the distance between them. In one case, I want to minimize the distance. In the other case, I want to maximize it.

#26: 1 July 2022

What is 1/6 ÷ 1/2?

In other words, how many halves are in one sixth? And in other colors, how many brown pieces fit in one olive piece?

#27: 2 July 2022

1. The area of a green square is what fraction of the area of the entire figure?

2. If you were to completely fill a green square with blue squares, what fraction of the area of the entire figure would those blue squares represent?

3. If you were to completely fill a green square with red squares, what fraction of the area of the entire figure would those red squares represent?

The 3 answers are equivalent fractions.

#28: 3 July 2022

If n is a whole number (1, 2, 3, ...), then the sum of the areas of the pink regions in n triangles is what fraction of the entire area of n triangles?

I began by finding the answer for n = 1, then for n = 2, then for n = 3...

#29: 4 July 2022

What fractions fill in these blanks?

The daisies in the blue rectangle are _____ of the daisies in the green rectangle, which are _____ of the daisies in the orange rectangle, which are _____ of the daisies in the entire arrangement of daisies.

The product of these three fractions (i.e., what you get when you multiply them together) completes this sentence: The daisies in the blue rectangle are _____ of the entire arrangement.

#30: 5 July 2022

The fractions indicate what part each shape is of the entire width or height of this square. For example, a blue rectangle is 2/3 of the height of the square and 1/6 of its width.

If the entire square is 12 cm on a side, what is the area of the red rectangle? [The area of a rectangle is the product of its height and width.]

The art of Piet Mondrian is a rich and popular context for mathematical thinking and reasoning.

#31: 6 July 2022

Fill in the boxes in the equation so that:

1) The sum of the numbers in the red boxes equals the number in the green box, and

2) the equation makes a true statement about the three regions of the divided polygon.

#32: 7 July 2022

This is a 3 × 4 × 5 rectangular prism. Its volume is 3 × 4 × 5 = 60 cubic units. A rectangular prism is cut from this one whose dimensions are (in order) 2/3, 3/4, and 4/5 of the dimensions of this one.

1. The volume of the new prism is what fraction of the volume of this one?

2. What is the product of the 3 fractions in the problem statement?

#33: 8 July 2022

#34: 9 July 2022

The fractions in each square are equal to the sum of the fractions in each of the vertices adjacent to them. For example, 1/2 is equal to fraction A + fraction B. Find fractions A, B, and C.

Find the sum of the fractions at the vertices; then find the sum of the fractions in the boxes. Explain why one sum is twice the other.

#35: 10 July 2022

If the blue region of this square equals 4/5, what does 1 look like?

I know what 4/5 looks like, so I began by figuring out what 1/5 looks like.

#36: 11 July 2022

Almonds make up two-thirds of a well-mixed bowl of almonds and cranberries. If half the mixture is removed and replaced with almonds, what fraction of the bowl will be almonds?

The original author of a problem similar to this one is unknown to me. That one features peanuts and raisins.

I began by illustrating the bowl as a rectangle partitioned (divided) equally into 3 parts. That allowed me to show that 2/3 of the bowl contains almonds.

#37: 12 July 2022

Use the figure to find 3 unique unit fractions whose sum is 1.

A unit fraction is a fraction with a numerator of 1. Your task is to find three different ones that add up to 1.

#38: 13 July 2022

What would it look like to shade 4/5 of each of these three models?

#39: 14 July 2022

What do you think? Is that equation true? Whatever your answer is, what makes you think so?

#40: 15 July 2022

The first pictorial equation shows that 1/2 (the first rectangle) of 2/5 (the second rectangle) is equal to 2/10 (the region in the third rectangle that's been shaded twice, once for each factor). This is what the symbolic equation says, too: One-half of two-fifths = 1/2 × 2/5 = 2/10.

Use this same way of illustrating fraction multiplication to figure out what the two missing figures could look like in the second pictorial equation.

#41: 16 July 2022

This rug is 2 1/2 feet wide by 1 1/3 feet long. What is its area?

The area of a rectangle is the product of its length and width (L × W).

I used the grid to decompose (break apart) the rug into 6 parts.

#42: 17 July 2022

This is a Tres Leches cake split into 5 pieces. There are 2 small, 2 medium, and 1 large pieces. As shown in the figure, it takes 4 tablespoons of sugar to make a large piece. How many small pieces can be made with 1 cup of sugar (assuming you have all the other ingredients you need)? [1 cup = 16 tbsp]

#43: 18 July 2022

The shaded part of this figure represents 3 2/3. Draw a picture of what 1 looks like.

We can't assume that each small square represents 1/4 unless we're told that a larger square is 1.

#44: 19 July 2022

Find the precise location of 2 1/3 /5 (“2 and 1/3 fifths”) yards on this number line.

#45: 20 July 2022

If the sum of two numbers, a and b, is 1, and their quotient is also 1, what is their product?

#46: 21 July 2022

This figure shows where the procedure for finding equivalent fractions comes from: Multiplying the numerator and denominator by the same non-zero number produces an equivalent fraction. For example, multiplying the numerator by 2 means multiplying the number of shaded parts by 2: The single red rectangle becomes the two blue rectangles. Multiplying the denominator by 2 means multiplying the total number of parts by 2: The three red rectangles become 6 blue rectangles. In numerical form, this looks like: 1/3 = (1 × 2)/(3 × 2) = 2/6. And 2/6 = 4/12.

Fill in the empty boxes in the second equation to produce fractions equivalent to 3/5.

The numbers that go in the boxes might be fractions, too. :)

#47: 22 July 2022

There are numbers that can fill in these boxes to form a pretty suprising relationship between the difference between two fractions and their product. When those numbers fill in the boxes, they form a pattern across all of the equations. What are those numbers?

#48: 23 July 2022

Fill in the blank to make a true statement that completes this lovely little math poem. Then recite it over and over and over. You found yourself singing it, right? Math'll do that to you.

Shoutout to @sbagley for sharing this favorite thing of his with us.

#49: 24 July 2022

This number line is drawn to scale. It shows the locations of five points. These five points are all positive. Fill in each blank with one of the four fractions to make equations that describe the relative locations of each of the points.

For example, filling in the blank in the first equation answers the question, "T is what fraction of S?" You don't need to find the values of any of the variables, but you could assign a value to one of them (e.g., T = 100) to help you.

P.S. I was surprised to learn that "maths" is actually not the plural of "math." It's a 'mass noun,' which is an uncountable noun like juice, music, or furniture.

The first equation asks, "T is what fraction of S"? I began by determining whether T is greater than S (in which case the fraction would be greater than 1), or if T is less than S (in which case the fraction would be less than 1).

#50: 25 July 2022

Let's say that the notation |A| means the size of set A. In this problem, |A| = 4.

What is the product (i.e., multiply) of the four fractions that answer these questions?

• |A| is what fraction of |B|?

• |B| is what fraction of |C|?

• |C| is what fraction of |D|?

• |D| is what fraction of |A|?

#51: 26 July 2022

Theo, Vanessa, and Rudy each thought about this division problem in a different way. For each of them, decide whether or not their reasoning is correct.

• Theo said: “2/3 is twice as big as 1/3, and I know that 1/3 goes into 1 three times. So 2/3 goes into 1 half as many times. The answer is 1 1/2.”

• Vanessa wrote: “1 = 3 (1/3's) = 1 1/2 (2/3's)”

• Rudy drew two rectangles, divided each rectangle into thirds, and shaded each pair of thirds in a different color. He used his drawings to conclude that “2 ÷ 2/3 = 3, so 1 ÷ 2/3 = 1 1/2.”

This task is authored by Susan Lamon on p. 13^2 of Teaching Fractions and Ratios for Understanding.

#52: 27 July 2022

If some people are sharing pizza and one person's share is 3/5 of a pizza, that means that 3 pizzas were shared among among 5 people. It could also be the case that 6 pizzas were shared among 10 people (because 3/5 = 6/10). In any case, there would be 3 pizzas for every 5 people. This way of thinking can help you solve this problem:

Suppose that 12 people ordered some pizzas. One person's share was 2/3 cheese pizza and 1/4 veggie pizza. How many pizzas of each type were ordered?

I know that if one person's share was 2/3 pizza, then 2 pizzas were shared among 3 people. If that's the case, how many pizzas were shared among all 12 people?

#53: 28 July 2022

I found this figure on the web. It shows how to convert a mixed number into an improper fraction.

Can you explain why it works? What does multiplying the whole number by the denominator tell us? What does the resulting '12' refer to? 12 what?

Why do we then add the numerator”? What is there 1 of?

Finally, why do we keep the same denominator?

If you had some other way to convert 4 1/3 to 13/3, that might help you make sense of this non-sense.

#54: 29 July 2022

Consider this:

If you start at home and head out to work, you must first cover 1/2 the total distance to get there. Then you must cover 1/2 of the remaining distance, which is 1/2 × 1/2, or 1/4 of the total distance. Then you must cover 1/2 of the remaining distance, which is 1/2 × 1/4, or 1/8 of the total distance. And so on ...

At this rate, how long will it take you to get to work?

#55: 30 July 2022

Roxy used a copier to reduce a picture to 4/5 of its original size and discarded the original. Later she learned that she was supposed to enlarge the original by 1/5, not reduce it by 1/5, but now she no longer has the original. How should Roxy set up the copier to produce a new copy from the reduced one that's 1/5 larger than the original?

This task is a slight variation of one authored by Susan Lamon on p. 197 of Teaching Fractions and Ratios for Understanding.

Once I can write the equation I need to solve, I'm good. So what I need to do is write the equation that models this question: What fraction of 4/5 of the original size (which is 100% or 1) produces a fraction that is 1/5 greater than the original size?

#56: 31 July 2022

For each figure, the bold lines show what the whole (or 1) looks like. For instance, the figure on the top left represents 1 1/2 shaded regions out of 2 1/2 wholes, or 1 1/2 / 2 1/2.

Find the fraction that names the shaded area in the other three figures. Then determine what all 4 fraction values have in common.

Yet another rich task from Susan Lamon's Teaching Fractions and Ratios for Understanding.

#57: 1 August 2022

"Quarter-biscuits" are a popular treat at the day care where Eli works. When Eli prepares quarter-biscuits from 3 whole biscuits and shares them equally among 8 children, how much of a whole biscuit does each kid get?

#58: 2 August 2022

Which one doesn't belong?

There are loads of "Which one doesn't belong?" (WODB) problems at wodb.ca.

#59: 3 August 2022

1. Suppose n is a positive whole number. As n gets bigger and bigger and bigger (e.g., 10, 100, 1000, ...), what value is 1/n getting closer and closer and closer to? That's precisely what this expression is asking for: "What is the limit as n approaches infinity of 1/n ?"

2. Now suppose n is a fraction between 0 and 1. As n gets smaller and smaller and smaller (e.g., 1/10, 1/100, 1/1000, ...), what happens to 1/n?

#60: 4 August 2022

The figure on the left shows that there are three fractions whose sum is 1 and whose numerators are 1, 2, and 3. The figure on the right represents another set of three fractions whose sum is 1 and whose numerators are also 1, 2, and 3. What are the denominators of those three fractions?

#61: 5 August 2022

I've heard it said that "dividing by a fraction is the same as multiplying by the reciprocal." As the figure demonstrates, this isn't true. Division and multiplication have different meanings. The figure on the left can help us answer the division question, "How many 1/6s go into 2/3?" In contrast, the figure on the right can help us answer the multiplication question, "What is 2/3 of 6/1?"

So now that we've established that "dividing by a fraction IS NOT the same as multiplying by the reciprocal," what IS the same about these two expressions: 2/3 ÷ 1/6 and 2/3 × 6/1? Use the figures to evaluate these expressions and find out.

#62: 6 August 2022

A. Where can you see '1/2' in the figure? Find the value of the reciprocal of 1/2, which is 1/(1/2), or 1 ÷ 1/2. What does that value refer to?

B. Where can you see '1/3' in the figure? Find the value of the reciprocal of 1/3, which is 1/(1/3), or 1 ÷ 1/3. What does that value refer to?

C. Where can you see '1/6' in the figure? Find the value of the reciprocal of 1/6, which is 1/(1/6), or 1 ÷ 1/6. What does that value refer to?

D. If 1/1,000,000 of a figure is orange, how many orange pieces would fit into the figure?

#63: 7 August 2022

1. The part of the circular clock formed by the hour and minute hands (marked in red) at 4 o'clock is what fraction of the entire circle?

2. Given that there are 360 degrees in a circle (why?), what is the measure of the angle formed by the hour and minute hands.

The measure of the angle is a fraction of 360.

#64: 8 August 2022

Express the length of arc BC (that purple length) as a fraction of the circumference of this circle (i.e., arc length / circumference). [The circumference of a circle is the distance around the circle.] That fraction tells us about the length of arc BC in relation to the circumference of the entire circle.

The reciprocal of that fraction tells us how many arcs of size BC fit along the entire circumference of the circle. [The reciprocal of a fraction a/b is the fraction b/a (assuming neither a nor b is 0).] Find the reciprocal of that fraction and then use a calculator to convert the fraction into a decimal. Do you recognize that decimal?

#65: 9 August 2022

A community clinic administers 100 doses of a vaccine every week. In order to meet a sudden demand, they reduce the size of a dose to 1/5 its original volume while still ensuring its effectiveness. As a result, how many doses of the vaccine will the clinic produce from 100 doses?

#66: 10 August 2022

3/2 gallons of paint are used to cover 2/3 of a wall to prepare it for a mural. How many more gallons of paint are needed to completely cover the wall?

#67: 11 August 2022

This is a matching problem that demonstrates the commutative property of multiplication using area models for fractions. The commutative property for multiplication says that the order in which we multiply factors doesn't change the product. Match each area model to an expression.

This comes from p. 42 of Cathy Fosnot and Maarten Dolk's Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents.

#68: 12 August 2022

In terms of how the colors share the area in each triangle, which of these is not like the others?

#69: 13 August 2022

The pink area is what fraction of the blue area?

Be careful here. The question isn't, "The pink area is what fraction of the outer square?"

#70: 14 August 2022

Which box of cat food is the best buy?

In the spirit of fractles, you're encouraged to do your work in fractions and not in decimals.

#71: 15 August 2022

This array represents the product 3/2 × 4/3. 1 is represented by how many squares in the array?

This is a slight variation of a problem on p. 102 of Fosnot and Dolk's (2002) Young Mathematicians at Work: Constructing Fractions, Decimals, and Percents.

I can find the part of the array that's equal to 1 by finding a 1 by 1 rectangle within the array. That's because 1 × 1 = 1 .

#72: 16 August 2022

The ants tell me that a race is 10 meters around this track. The distance between each mark on the track is 1/3 of a meter. How many laps are there in an ant race?

#73: 17 August 2022

Each of these figures is labeled with what part of a whole (i.e., what fraction of 1) the figure represents. For example, the entire triangle is 1/10 of a whole.

What part of a whole does the yellow region in each figure represent?

## If you want to be sure you've interpreted the question as I intended, click here to see my strategy in the case of the triangle.

The yellow region in the triangle is 1/6 of 1/10 of the whole.

#74: 18 August 2022

Brittany makes 3 out of 6 shots at the free-throw line during practice. The next day, she makes 5 out of 6 shots. That means she made 8 out of 12 shots in those two days, right? But I thought you needed common denominators when you add. What happened? Is 8 out of 12 *not* the right answer? What do you think?

#75: 19 August 2022

This is the "Flying Geese" quilt design, which was used along the Underground Railroad to secretly provide directions to enslaved people on the run and fleeing enslavement.

This design is just one of the panels in a memorial quilt containing many different "secret code" designs, all the same size. If the blue region is 1/40 of the area of that quilt, how many panels does the quilt contain?

## Click here if you'd like a recommendation for a fractle you may want to solve before solving this one.

I suggest completing the fractle #73 from August 17, 2022 before completing this one.

#76: 20 August 2022

This is an input-output machine. What fraction completes this equation: output = _________ × input ?

#77: 21 August 2022

Emma's (green) share of this cake is what fraction of Mika's (pink) share?

#78: 22 August 2022

This is the flag of the Republic of Seychelles, which is comprised of 115 islands that lie in the Indian Ocean east of Africa. What fraction of this flag is neither blue nor green? [The area of a triangle is 1/2(base × height). That's because a triangle is 1/2 of a parallelogram, and the area of a paralleogram is base × height.]

#79: 23 August 2022

What is 1/5 ÷ 2/3?

This is a tough one because the divisor, 2/3, is greater than the dividend, 1/5. I've drawn a picture that can help you get started.

You're encouraged to use the picture to answer the question, not that procedure we all "learned" in school that never made any sense to us. Instead, rely on your visual sense. It's reliable.

#80: 24 August 2022

We can use this figure to understand the concept of multiplicative inverses.

Fill in the blanks: If there are _____ equal-sized pieces in a whole, then each piece is _____ of the whole.

Now find two fractions whose product is 1: _____ × _____

These fractions are multiplicative inverses of each other. 🤓

#81: 25 August 2022

Each of these figures is labeled with a number. For each figure, answer this question: That number contains how many pieces that are equal in area to the green region?

#82: 26 August 2022

1/4 of a mixture contains orange candy. 1/3 of the mixture that's not orange contains lemon candy. 1/2 of the mixture that's neither orange nor lemon contains strawberry candy. And 1/2 of the mixture that's neither, orange, lemon, or strawberry contains grape candy.

Divide this rectangle to show what fraction of the mixture contains each flavor of candy. Then determine how much of the mixture, at most, could be blueberry.

#83: 27 August 2022

Use the bottom row of figures to determine what fraction of the top hexagon is shaded.

#84: 28 August 2022

A ratio is a relationship between two quantities, and a proportion is a statement that two ratios are equal. You probably "learned" in school that you can solve a proportion (in this case, to find a ratio equivalent to 3/6 with a denominator of 4) by cross-multiplying. But why does that work? Where do the original denominators go from that first step to the second? Don't we have to multiply each side of an equation by the same value to maintain the equality? We didn't do that here.

Fill in the boxes to find fractions equivalent to 3/6 and x/4. Then see if you can determine what we actually multiply each side of a proportion equation by when we cross-multiply.

#85: 29 August 2022

Write an expression that means "a copies of b." Then write a second expression that means "a things shared among b people." These expressions can help you answer today's fractle:

Find values of a and b so that these problems have the same answer:

• If a pieces of Albert's Fruit Chews are shared among b people, how many pieces does each person get?

• If there are a pieces of Albert's Fruit Chews in each bag and I have b bags, how many pieces of candy do I have?

#86: 30 August 2022

This is called a complex fraction. What do you make of it? What's it equal to? Try to figure it out by reasoning to a solution rather than using a procedure that isn't meaningful to you. "Reasoning it out" means making justified claims of the form, "I know ______ is true, because ______ .

#87: 31 August 2022

Use the lines on this figure to show how the pattern of triangles can be used to divide the square into two halves, three thirds, six sixths and nine ninths. You can do your work on this digital applet or by downloading a PDF of a bunch of these figures.

This one comes from the website NRICH at the University of Cambridge.

#88: 1 September 2022

These are called continued fractions. If a and b are positive, and a < b, then which fraction is greater?

You might want to get pencil and paper for this one. There's a lot of reasoning required.

This one comes from the website NRICH at the University of Cambridge.

#89: 2 September 2022

These are Cuisenaire rods. They're great for learning fractions because they're not already partitioned like fraction strips are.

• The red rod is what fraction of the black rod?

• The yellow rod is what fraction of the red rod?

• If the white rod is 1/9, the sum of what three rods of the same color is equal to 2?

#90: 3 September 2022

One-fourth of two-thirds of the orange splat's area is equal to two-thirds of three-fourths of the blue splat's area.

The orange splat's area is how many times the size of the blue splat's area?