Name the whole before the fraction

d1. A 12 cm strip and an 8 cm strip are each folded in half. Are the two halves the same length? Explain.

Answer: No. The halves are 6 cm and 4 cm because the wholes have different lengths.

Look for:

• States that the halves differ.
• Connects the difference to the whole lengths.

Teacher response: Likely difficulty: Treating 1/2 as one fixed physical size. Response: Overlay the two whole strips, then compare one half of each without stretching either strip.

d2. A strip has four equal parts and three are shaded. What is the whole, and what fraction is shaded?

Answer: The entire four-part strip is the whole; 3/4 is shaded.

Look for:

• Names all four parts as one whole.
• Names 3/4 of that whole.

Teacher response: Likely difficulty: Calling only the shaded region the whole. Response: Trace all four parts with one loop before counting the shaded parts.

d3. One ribbon is cut into three unequal lengths. Is every piece 1/3 of the ribbon? Why?

Answer: No. Thirds must be three equal-sized parts of the same whole.

Look for:

• Rejects the one-third claim.
• Uses equal size as the reason.

Teacher response: Likely difficulty: Counting pieces without checking their size. Response: Align the pieces at one end and ask which piece contains more ribbon.

w1. Mia shades 1/2 of a 12 cm strip. Sam shades 1/2 of an 8 cm strip. Compare their shaded lengths.

1. Label each entire strip as one whole.
2. Divide each whole into two equal lengths.
3. Measure a half of each: 12 ÷ 2 = 6 and 8 ÷ 2 = 4.
4. Compare the lengths using the original whole labels.

Worked example: Mia shades 6 cm; Sam shades 4 cm. The fraction names match, but the amounts differ.

Look for:

• Identifies both reference wholes.
• Explains 6 cm versus 4 cm.

Teacher response: Likely difficulty: Assuming equal fraction names guarantee equal amounts. Response: Ask the learner to finish: “One half of ___ is ___.” Keep the object in the sentence.

w2. An 8-square rectangle is one whole. A student shades 2 squares and calls it 2/2. Repair the label.

1. Outline all 8 equal squares as the whole.
2. Use 8 as the number of equal parts in that whole.
3. Count the 2 shaded parts for the numerator.
4. Group the rectangle into four equal groups of 2 to see 1/4.

Worked example: 2/8, or 1/4, of the whole is shaded.

Look for:

• Uses 8 equal parts in the denominator.
• Accepts 2/8 or a justified 1/4.

Teacher response: Likely difficulty: Using the number of shaded parts as the denominator. Response: Cover the shading briefly and ask how many equal squares still make the whole.

p1. Fold a strip into four equal parts. Shade one part and outline the whole. Name the shaded fraction.

Answer: 1/4 of the entire strip.

Look for:

• Outlines four equal parts as one whole.
• Labels 1/4.

Teacher response: Likely difficulty: Naming the shaded piece as one whole. Response: Point to the four-part outline and ask how many copies of the shaded piece fill it.

p2. Two same-sized rectangles are each one whole. One shows 2/4 shaded; the other shows 1/2. Compare the shaded amounts.

Answer: They are equal because two fourths fill one half of the same-sized whole.

Look for:

• States equal amounts.
• Refers to equal-sized wholes and equal coverage.

Teacher response: Likely difficulty: Comparing only the number of shaded pieces. Response: Place the half over the two fourths and compare their boundaries.

p3. A small cake and a large cake each have 1/4 left. Can you tell whether the remaining amounts are equal from the fraction labels alone?

Answer: No. The whole cake sizes are needed; equal fraction labels alone do not establish equal amounts.

Look for:

• Says the labels alone are insufficient.
• Names the missing whole-size information.

Teacher response: Likely difficulty: Ignoring the sizes of the reference cakes. Response: Draw two visibly different cakes and shade a fourth of each.

p4. A 16 cm ribbon is one whole. Four equal sections are marked. How long is 1/4 of this whole?

Answer: 4 cm.

Look for:

• Finds 4 cm.
• Checks four equal lengths make 16 cm.

Teacher response: Likely difficulty: Using 4 as both the number and length of parts without a partition argument. Response: Place four equal segments end to end and check that their total length is 16 cm.

p5. A tray has 8 equal compartments. The whole tray is one whole and 2 compartments are filled. Name the filled fraction.

Answer: 2/8, equivalent to 1/4.

Look for:

• Uses all 8 compartments.
• Names 2/8 or justified 1/4.

Teacher response: Likely difficulty: Counting only the six empty compartments for the denominator. Response: Count every compartment in the tray, including the filled ones.

p6. Draw two different-sized rectangles. Shade 1/2 of each. Write one sentence about their fraction names and one about their shaded areas.

Answer: Both show 1/2 of their own whole; the larger rectangle has the larger shaded area.

Look for:

• Partitions each rectangle into equal halves.
• Distinguishes same fraction from different area.

Teacher response: Likely difficulty: Equating fraction equality with area equality across different wholes. Response: Compare the entire rectangles before comparing the shaded regions.

t1. A poster says: “Lee ate 1/2 of a sandwich; Noor ate 1/4, so Lee ate more.” What information must be checked before agreeing?

Answer: Check that the whole sandwiches have the same size, or find their actual sizes. The fraction labels alone do not settle the amounts eaten.

Look for:

• Identifies whole sandwich size as necessary.
• Explains why fraction labels alone are insufficient.

Teacher response: Likely difficulty: Comparing fractions as amounts across unstated wholes. Response: Ask the student to draw a sandwich for which a fourth is larger than a half of a much smaller sandwich.

t2. A 6 cm piece is 1/2 of one ribbon and 1/3 of another. How long is each whole ribbon? Explain with copies of the piece.

Answer: The first ribbon is 12 cm; the second is 18 cm. Two or three copies of the 6 cm piece reconstruct the respective wholes.

Look for:

• Builds 12 cm from two copies.
• Builds 18 cm from three copies and explains different wholes.

Teacher response: Likely difficulty: Assuming the same physical part requires the same fraction name. Response: Build each whole by repeating the same 6 cm piece the stated number of times.