Equal Parts Workshop: teaching companion

prepare. Before starting: Recognize a whole and count equal parts.

materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).

activity-1. Fold a paper rectangle in halves, then fourths. Open it and trace the folds.

activity-2. Draw a second rectangle split at 1/4 of its width. Compare the two piece sizes.

activity-3. Partners sort the drawings into equal and unequal parts. Give a reason for each choice.

phase-1. 5 min · warmup: Look at the first model or task. What is the whole? What do you notice?

Answer: Students: Sketch an initial prediction and explain it to a partner.

phase-2. 8 min · model: Fold a paper rectangle in halves, then fourths. Open it and trace the folds.

Answer: Students: Follow the model; label the whole and the equal parts.

phase-3. 10 min · partner: Draw a second rectangle split at 1/4 of its width. Compare the two piece sizes.

Answer: Students: Work in pairs. One models, the other checks; switch roles.

phase-4. 7 min · discussion: Partners sort the drawings into equal and unequal parts. Give a reason for each choice.

Answer: Students: Compare two solutions and explain the relationship.

phase-5. 10 min · practice: Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.

Answer: Students: Solve independently, then compare the reasoning.

phase-6. 5 min · exit: Three unequal pieces are each one third. Is that correct? Explain the mistake in calling any one of four unequal pieces one fourth.

Answer: Students: Complete the two exit questions independently.

d1. 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?

Answer: No. The four pieces have unequal areas, so the count alone does not establish fourths.

Teacher response: If the learner calls all four pieces fourths because there are four. Keep the 8 cm whole fixed. Mark four 2 cm widths, then ask what property the repaired pieces share.

d2. A square is cut along a diagonal into two triangles. Is each triangle one half even though neither looks like a rectangular strip?

Answer: Yes. The two equal-area triangles fill the same square.

Teacher response: If the learner rejects halves because the pieces are triangular. Fold one triangle onto the other, then separate equal area from a preferred piece shape.

r1. A whole has three pieces; two are small and one is large. What must change before all three can be called thirds?

Answer: The pieces must have equal area while together filling the same whole.

e1. Three unequal pieces are each one third. Is that correct? Choices: Yes / No.

Answer: No

Teacher response: 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.

e2. Explain the mistake in calling any one of four unequal pieces one fourth.

Answer: The count is four but the areas are not equal.

Look for:

• Identify that having four pieces gives only a count.
• State that the four pieces have unequal areas.
• Explain that one fourth requires one of four equal-area parts of the whole.

Teacher response: 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.

teach-1. Outline an 8 cm strip and label it one whole.

teach-2. Compare the unequal four-piece partition with four equal 2 cm pieces; ask the learner to match areas by overlaying.

teach-3. Have the learner repair a fresh three-piece strip and explain why a piece can now be called one third.

next-1. 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.

Teacher response: Continue to the linked next lesson. Build and repeat a unit fraction after the learner can verify equal parts. Next original lesson: /lessons/unit-fraction-builders/.

next-2. e1 does not yet meet this criterion: e1 answers “No” to the claim that three unequal pieces are each one third.

Teacher response: 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. Short intervention: /interventions/whole-size/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains that four pieces are fourths only when their areas are equal.

Teacher response: Ask the learner to point to the unequal areas in e2, repair the partition, and explain the equal-area requirement before retrying the explanation. Short intervention: /interventions/whole-size/.

c1. Are the parts in this strip equal in size? Choices: Equal parts / Unequal parts. Strip model: 1 same-sized whole, each divided into 2 equal parts; 1 parts are shaded altogether.

Answer: Equal parts

Teacher response: 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.

c2. Are the parts in this strip equal in size? Choices: Equal parts / Unequal parts. Strip model: one whole is divided into pieces with relative widths 25%, 75%.

Answer: Unequal parts

Teacher response: 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.

c3. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 3 equal parts; 1 parts are shaded altogether.

Answer: 1/3

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

c4. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 4 equal parts; 1 parts are shaded altogether.

Answer: 1/4

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

c5. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 4 equal parts; 2 parts are shaded altogether.

Answer: 2/4

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

c6. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 6 equal parts; 1 parts are shaded altogether.

Answer: 1/6

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

c7. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 8 equal parts; 1 parts are shaded altogether.

Answer: 1/8

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

c8. A rectangle has four pieces, all with the same area. Is each piece one fourth? Choices: Yes / No.

Answer: Yes

Teacher response: 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.

s1. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 2 equal parts; 1 parts are shaded altogether.

Answer: 1/2

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

s2. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 3 equal parts; 1 parts are shaded altogether.

Answer: 1/3

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

s3. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 4 equal parts; 1 parts are shaded altogether.

Answer: 1/4

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

s4. What fraction is shaded? Write its value. Strip model: 1 same-sized whole, each divided into 4 equal parts; 2 parts are shaded altogether.

Answer: 2/4

Teacher response: Treats the denominator as a count of selected pieces instead of the number of equal parts in one whole. 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.

h1. Draw a four-part rectangle that does not show fourths. Explain.

Answer: Three narrow pieces and one wide piece are not fourths because the areas differ.

Look for:

• Draw one rectangle partitioned into exactly four pieces.
• Make at least two piece areas visibly unequal.
• Explain that four pieces are fourths only when their areas are equal.

Teacher response: 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.

h2. Two equal-area pieces have different shapes. Can each be a half?

Answer: Yes. Equal area matters, even when shapes differ.

Look for:

• Answer that differently shaped pieces can each be one half.
• Require equal areas and a shared whole.
• Explain or sketch how the two pieces together fill that whole.

Teacher response: 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.

h3. Repair an unequal two-part strip without changing its whole.

Answer: Move the partition to the midpoint so both parts have half the area.

Look for:

• Keep the original strip length and whole fixed.
• Move or redraw the cut to create two equal-area parts.
• Identify each repaired part as one half and justify its equal size.

Teacher response: 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.

h4. How could a partner verify that your six pieces are sixths?

Answer: Compare their areas or fold them onto each other; six equal pieces fill the whole.

Look for:

• Check that exactly six pieces fill the whole with no gaps or overlaps.
• Describe a way to compare their areas, such as folding or overlaying.
• Conclude that each piece is 1/6 only if all six areas match.

Teacher response: 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.

e1. Three unequal pieces are each one third. Is that correct? Choices: Yes / No.

Answer: No

Teacher response: 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.

e2. Explain the mistake in calling any one of four unequal pieces one fourth.

Answer: The count is four but the areas are not equal.

Look for:

• Identify that having four pieces gives only a count.
• State that the four pieces have unequal areas.
• Explain that one fourth requires one of four equal-area parts of the whole.

Teacher response: 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.