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

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?

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?

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?

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 ?

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?

This one's a classic ...

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

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

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.

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?

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 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.

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.

Without drawing any additional lines, shade 2/9 of this rectangle.

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.

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.

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

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]

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

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.

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?

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.

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.

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|?

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.

"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?

Which one doesn't belong?

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

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?

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?

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.

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?

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.

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?

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?"

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?

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

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.

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?

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.

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.

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?

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?