Distinguish equal-size parts from unequal parts before naming a fraction.
Teaching companion · Version 2.0.0
Establish equal area before naming halves, thirds, or fourths.
The original lesson PDFs remain version 1.0.0. This supplement adds diagnostic probes, teaching responses, and question-specific rubrics. Use the supplement below for the updated guidance.
Ask for an explanation or drawing before choosing a response. A wrong answer alone does not identify its cause.
A strip is cut into four pieces with widths 1, 1, 1, and 5 cm. Can every piece be called one fourth of the strip?
Look for: No. The four pieces have unequal areas, so the count alone does not establish fourths.
If you observe: Calls all four pieces fourths because there are four.
Respond: Keep the 8 cm whole fixed. Mark four 2 cm widths, then ask what property the repaired pieces share.
A square is cut along a diagonal into two triangles. Is each triangle one half even though neither looks like a rectangular strip?
Look for: Yes. The two equal-area triangles fill the same square.
If you observe: Rejects halves because the pieces are triangular.
Respond: Fold one triangle onto the other, then separate equal area from a preferred piece shape.
Check independently: A whole has three pieces; two are small and one is large. What must change before all three can be called thirds?
The pieces must have equal area while together filling the same whole.
Use the whole size short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
e1 meets its criterion: e1 answers “No” to the claim that three unequal pieces are each one third. e2 also meets its criterion: e2 explains that four pieces are fourths only when their areas are equal.
Continue to the linked next lesson. Build and repeat a unit fraction after the learner can verify equal parts.
Next lesson: Unit Fraction Builderse1 does not yet meet this criterion: e1 answers “No” to the claim that three unequal pieces are each one third.
Ask the learner to explain the attempt, then use the matching diagnostic probe to identify the difficulty. Work through the focused reteach below and its check. Use the linked short intervention if the difficulty persists; retry e1 and e2 before moving on.
Open the linked short interventione1 meets its criterion, but e2 does not yet show this evidence: e2 explains that four pieces are fourths only when their areas are equal.
Ask the learner to point to the unequal areas in e2, repair the partition, and explain the equal-area requirement before retrying the explanation.
Open the linked short interventionUse these as follow-up prompts, then examine the learner’s model or explanation. Open responses may differ from the sample while satisfying the criteria.
Strip model: 1 same-sized whole, each divided into 2 equal parts; 1 parts are shaded altogether.
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Strip model: one whole is divided into pieces with relative widths 25%, 75%.
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Strip model: 1 same-sized whole, each divided into 3 equal parts; 1 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Strip model: 1 same-sized whole, each divided into 4 equal parts; 1 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Strip model: 1 same-sized whole, each divided into 4 equal parts; 2 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Strip model: 1 same-sized whole, each divided into 6 equal parts; 1 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Strip model: 1 same-sized whole, each divided into 8 equal parts; 1 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Strip model: 1 same-sized whole, each divided into 2 equal parts; 1 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Strip model: 1 same-sized whole, each divided into 3 equal parts; 1 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Strip model: 1 same-sized whole, each divided into 4 equal parts; 1 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Strip model: 1 same-sized whole, each divided into 4 equal parts; 2 parts are shaded altogether.
Watch for: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole.
Try: Outline one whole first. Count all the equal pieces that fill it, label one piece 1/b, then repeat that piece until the whole is covered with no gaps.
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Look for in the response:
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Look for in the response:
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Look for in the response:
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Look for in the response:
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Watch for: Counts pieces without checking equal area.
Try: Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Look for in the response:
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.
Eight core questions and a two-question exit ticket. Print one set per student.
Choices: Equal parts / Unequal parts
Choices: Equal parts / Unequal parts
Choices: Yes / No
Choices: Yes / No
Recognize a whole and count equal parts.
Look at the first model or task. What is the whole? What do you notice?
Fold a paper rectangle in halves, then fourths. Open it and trace the folds.
Draw a second rectangle split at 1/4 of its width. Compare the two piece sizes.
Partners sort the drawings into equal and unequal parts. Give a reason for each choice.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Three unequal pieces are each one third. Is that correct? Explain the mistake in calling any one of four unequal pieces one fourth.
Counts pieces without checking equal area.
Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition.
Teacher answer key. Open responses include a rubric and one valid sample.
Choices: Equal parts / Unequal parts
Each part has the same width.
Choices: Equal parts / Unequal parts
One part has width 1/4 and the other 3/4.
1 parts of size 1/3 represent 1/3.
1 parts of size 1/4 represent 1/4.
2 parts of size 1/4 represent 2/4.
1 parts of size 1/6 represent 1/6.
1 parts of size 1/8 represent 1/8.
Choices: Yes / No
Four equal areas make fourths.
1 parts of size 1/2 represent 1/2.
1 parts of size 1/3 represent 1/3.
1 parts of size 1/4 represent 1/4.
2 parts of size 1/4 represent 2/4.
Three narrow pieces and one wide piece are not fourths because the areas differ.
Yes. Equal area matters, even when shapes differ.
Move the partition to the midpoint so both parts have half the area.
Compare their areas or fold them onto each other; six equal pieces fill the whole.
Choices: Yes / No
Thirds must be equal in size.
The count is four but the areas are not equal.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
Choices: Yes / No
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
Choices: Yes / No
Designed for Grade 3. Distinguish equal-size parts from unequal parts before naming a fraction.
Version 1.0.0. Original, AI-assisted lesson content. Free local classroom use; public reuse terms have not yet been assigned.
These lessons have not yet been field-tested in classrooms. Use the exit ticket and student explanations to decide what to revisit.