Use two different-sized paper wholes to separate a fraction name from the amount it describes.
Listen for the idea
A fraction refers to a specified whole; amount comparisons need equal-sized wholes.
Identify the whole before naming its equal parts.
Explain why the same fraction of different wholes can be different amounts.
Materials: Paper strips, one 12 cm and one 8 cm long; Pencil and ruler.
A 12-minute route
2 minutes: Try the three diagnostic questions. Ask for a brief explanation.
3 minutes: Teach one worked example that addresses the difficulty.
5 minutes: Choose three practice tasks and discuss the reasoning.
2 minutes: Give one transfer task independently and decide the next step.
The second worked example, three unused practice tasks, and remaining transfer task are available for a follow-up session. Allow more time if the group needs to build the models.
Teaching move: Put both strips on the table before naming any fraction. Have the student trace the entire strip, then fold it into equal parts. Keep the whole visible beside every shaded part.
Use the answer notes to interpret what you hear. Choose a response for each question to get a suggested teaching step. These choices describe this brief check and are not a mastery score.
1 A 12 cm strip and an 8 cm strip are each folded in half. Are the two halves the same length? Explain.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: No. The halves are 6 cm and 4 cm because the wholes have different lengths.
Listen for this difficulty: Treating 1/2 as one fixed physical size.
Try this response: Overlay the two whole strips, then compare one half of each without stretching either strip.
Look for:
States that the halves differ.
Connects the difference to the whole lengths.
2 A strip has four equal parts and three are shaded. What is the whole, and what fraction is shaded?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: The entire four-part strip is the whole; 3/4 is shaded.
Listen for this difficulty: Calling only the shaded region the whole.
Try this response: Trace all four parts with one loop before counting the shaded parts.
Look for:
Names all four parts as one whole.
Names 3/4 of that whole.
3 One ribbon is cut into three unequal lengths. Is every piece 1/3 of the ribbon? Why?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: No. Thirds must be three equal-sized parts of the same whole.
Listen for this difficulty: Counting pieces without checking their size.
Try this response: Align the pieces at one end and ask which piece contains more ribbon.
Look for:
Rejects the one-third claim.
Uses equal size as the reason.
2 · Model together
Two worked examples
Use one now. Ask the learner to explain each step before revealing the next part of your drawing.
1 Mia shades 1/2 of a 12 cm strip. Sam shades 1/2 of an 8 cm strip. Compare their shaded lengths.
Label each entire strip as one whole.
Divide each whole into two equal lengths.
Measure a half of each: 12 ÷ 2 = 6 and 8 ÷ 2 = 4.
Compare the lengths using the original whole labels.
Result: Mia shades 6 cm; Sam shades 4 cm. The fraction names match, but the amounts differ.
Teaching notes
Listen for this difficulty: Assuming equal fraction names guarantee equal amounts.
Try this response: Ask the learner to finish: “One half of ___ is ___.” Keep the object in the sentence.
Look for:
Identifies both reference wholes.
Explains 6 cm versus 4 cm.
2 An 8-square rectangle is one whole. A student shades 2 squares and calls it 2/2. Repair the label.
Outline all 8 equal squares as the whole.
Use 8 as the number of equal parts in that whole.
Count the 2 shaded parts for the numerator.
Group the rectangle into four equal groups of 2 to see 1/4.
Result: 2/8, or 1/4, of the whole is shaded.
Teaching notes
Listen for this difficulty: Using the number of shaded parts as the denominator.
Try this response: Cover the shading briefly and ask how many equal squares still make the whole.
Look for:
Uses 8 equal parts in the denominator.
Accepts 2/8 or a justified 1/4.
3 · Guided practice
Choose three tasks
Mix a calculation or drawing with an explanation or error repair. Keep the other tasks for another short session.
1 Fold a strip into four equal parts. Shade one part and outline the whole. Name the shaded fraction.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/4 of the entire strip.
Listen for this difficulty: Naming the shaded piece as one whole.
Try this response: Point to the four-part outline and ask how many copies of the shaded piece fill it.
Look for:
Outlines four equal parts as one whole.
Labels 1/4.
2 Two same-sized rectangles are each one whole. One shows 2/4 shaded; the other shows 1/2. Compare the shaded amounts.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: They are equal because two fourths fill one half of the same-sized whole.
Listen for this difficulty: Comparing only the number of shaded pieces.
Try this response: Place the half over the two fourths and compare their boundaries.
Look for:
States equal amounts.
Refers to equal-sized wholes and equal coverage.
3 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?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: No. The whole cake sizes are needed; equal fraction labels alone do not establish equal amounts.
Listen for this difficulty: Ignoring the sizes of the reference cakes.
Try this response: Draw two visibly different cakes and shade a fourth of each.
Look for:
Says the labels alone are insufficient.
Names the missing whole-size information.
4 A 16 cm ribbon is one whole. Four equal sections are marked. How long is 1/4 of this whole?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 4 cm.
Listen for this difficulty: Using 4 as both the number and length of parts without a partition argument.
Try this response: Place four equal segments end to end and check that their total length is 16 cm.
Look for:
Finds 4 cm.
Checks four equal lengths make 16 cm.
5 A tray has 8 equal compartments. The whole tray is one whole and 2 compartments are filled. Name the filled fraction.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 2/8, equivalent to 1/4.
Listen for this difficulty: Counting only the six empty compartments for the denominator.
Try this response: Count every compartment in the tray, including the filled ones.
Look for:
Uses all 8 compartments.
Names 2/8 or justified 1/4.
6 Draw two different-sized rectangles. Shade 1/2 of each. Write one sentence about their fraction names and one about their shaded areas.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Both show 1/2 of their own whole; the larger rectangle has the larger shaded area.
Listen for this difficulty: Equating fraction equality with area equality across different wholes.
Try this response: Compare the entire rectangles before comparing the shaded regions.
Look for:
Partitions each rectangle into equal halves.
Distinguishes same fraction from different area.
4 · Independent transfer
Try a fresh situation
Choose one task without coaching. Check the explanation against both criteria in its teaching notes; use the other task for a later check.
1 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?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Check that the whole sandwiches have the same size, or find their actual sizes. The fraction labels alone do not settle the amounts eaten.
Listen for this difficulty: Comparing fractions as amounts across unstated wholes.
Try this response: Ask the student to draw a sandwich for which a fourth is larger than a half of a much smaller sandwich.
Look for:
Identifies whole sandwich size as necessary.
Explains why fraction labels alone are insufficient.
2 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.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: The first ribbon is 12 cm; the second is 18 cm. Two or three copies of the 6 cm piece reconstruct the respective wholes.
Listen for this difficulty: Assuming the same physical part requires the same fraction name.
Try this response: Build each whole by repeating the same 6 cm piece the stated number of times.
Look for:
Builds 12 cm from two copies.
Builds 18 cm from three copies and explains different wholes.
What to do next
If the explanation holds: Move on when the student names the reference whole in both transfer tasks and distinguishes fraction from physical amount.
If the difficulty remains: If the whole is omitted, return to one outlined strip. Rebuild that whole with equal copies of a single part before comparing two strips.
Standards: 3.NF.A.1, 3.NF.A.3.d. Grade 3 alignment describes the mathematical work in this pack.
The tasks, worked examples, and teacher responses are original Fraction Classroom materials, version 2.0.0. The teaching approach draws on noticing the unit, moving between representations, and responding to student reasoning in the linked Illustrative Mathematics lesson. Its text and exercises are not reproduced here.