Equivalent Strip Lab: 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. Draw three strips of exactly the same length. Shade one half of the first.

activity-2. Divide the second into fourths and the third into sixths. Shade to the same endpoint.

activity-3. Pair up to explain what changed and what stayed equal. Repeat with two thirds and sixths.

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: Draw three strips of exactly the same length. Shade one half of the first.

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

phase-3. 10 min · partner: Divide the second into fourths and the third into sixths. Shade to the same endpoint.

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

phase-4. 7 min · discussion: Pair up to explain what changed and what stayed equal. Repeat with two thirds and sixths.

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: Compare 3/6 and 1/2. Choose <, =, or >. The wholes are the same size. Explain why 1/2 and 2/4 are equivalent.

Answer: Students: Complete the two exit questions independently.

d1. Rename 2/3 in sixths. Explain what happens to each third.

Answer: 2/3 = 4/6; every third is split into two sixths.

Teacher response: If the learner writes 2/6 by changing only the denominator. Split each third of the same shaded strip in two and recount all six pieces and the four shaded pieces.

d2. A learner claims 2/3 = 3/4 because both numbers increase by one. Does the claim preserve the endpoint?

Answer: No. 2/3 = 8/12 while 3/4 = 9/12.

Teacher response: If the learner uses an additive rule for equivalent fractions. Use a twelfths strip to expose the different endpoints, then contrast with multiplying both counts by two.

r1. Rename 3/4 in twelfths and explain why its amount stays the same.

Answer: 3/4 = 9/12: each fourth becomes three twelfths, while the shaded endpoint stays fixed.

e1. Compare 3/6 and 1/2. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

e2. Explain why 1/2 and 2/4 are equivalent.

Answer: Repartition the same whole: each half becomes two fourths, and the shaded endpoint stays fixed.

Look for:

• Keep the whole fixed when moving from halves to fourths.
• Split each half into two equal fourths, giving 1/2 = 2/4.
• Explain that the shaded amount or endpoint stays unchanged.

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

teach-1. Shade half of a fixed strip and keep its boundary visible.

teach-2. Split every half into two fourths; recount two shaded parts out of four.

teach-3. Repeat with 2/3 becoming 4/6, recording the shared factor on both arrows.

next-1. e1 meets its criterion: e1 gives = for 3/6 compared with 1/2. e2 also meets its criterion: e2 explains 1/2 = 2/4 by splitting each half into two fourths of the same whole without moving the endpoint.

Teacher response: Continue to the linked next lesson. Use the repartitioning factor to solve missing numerators and denominators. Next original lesson: /lessons/missing-number-bridges/.

next-2. e1 does not yet meet this criterion: e1 gives = for 3/6 compared with 1/2.

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/equivalent-factors/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains 1/2 = 2/4 by splitting each half into two fourths of the same whole without moving the endpoint.

Teacher response: Ask the learner to demonstrate the specific half-to-fourths repartition in e2, naming what changes and what stays fixed. Short intervention: /interventions/equivalent-factors/.

c1. The strip shows 2/4. Enter the same amount as a fraction. Strip model: 1 same-sized whole, each divided into 4 equal parts; 2 parts are shaded altogether.

Answer: 1/2

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

c2. Compare 1/2 and 2/4. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

c3. Compare 2/3 and 4/6. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

c4. Rename 3/4 as an equivalent fraction with denominator 8. Strip model: 1 same-sized whole, each divided into 8 equal parts; 6 parts are shaded altogether.

Answer: 6/8

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

c5. Compare 3/6 and 1/2. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

c6. Compare 2/5 and 4/10. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

c7. Write the value shown by eight of twelve parts. Strip model: 1 same-sized whole, each divided into 12 equal parts; 8 parts are shaded altogether.

Answer: 2/3

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

c8. Compare 3/4 and 9/12. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

s1. What amount is shaded? Count two fourths. Strip model: 1 same-sized whole, each divided into 4 equal parts; 2 parts are shaded altogether.

Answer: 2/4

Teacher response: Reverses the shaded count and the total equal-part count. Use two labels: “equal pieces in one whole” under the bar and “pieces selected” above it. Point to every piece while naming the denominator, then only the selected pieces while naming the numerator.

s2. Compare 1/2 and 2/4. Choose <, =, or >. Split each half into two equal parts.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

s3. What amount is shaded? Count three sixths. Strip model: 1 same-sized whole, each divided into 6 equal parts; 3 parts are shaded altogether.

Answer: 3/6

Teacher response: Reverses the shaded count and the total equal-part count. Use two labels: “equal pieces in one whole” under the bar and “pieces selected” above it. Point to every piece while naming the denominator, then only the selected pieces while naming the numerator.

s4. Compare 1/3 and 2/6. Choose <, =, or >. Split each third into two equal parts.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

h1. Give three different names for 2/3 and explain the common endpoint.

Answer: 2/3, 4/6 and 8/12 cover the same share of equal-length strips.

Look for:

• Give three distinct fraction names equal to 2/3; 2/3 itself may be one of them.
• Show the same factor applied to numerator and denominator for each renaming.
• Explain the common endpoint using the same whole or unit line.

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

h2. A partner says 2/3 = 3/4 because both numbers increased by one. Give a counterexample.

Answer: Two thirds is 8/12 and three fourths is 9/12, so their endpoints differ.

Look for:

• Reject 2/3 = 3/4.
• Provide evidence of different values, such as 8/12 versus 9/12.
• Explain that adding one to both counts does not preserve the fraction as multiplying both by the same factor does.

Teacher response: Adds the same number to numerator and denominator and claims the fraction stays equal. Put 2/3 and 3/4 on the same twelfths strip: 8/12 and 9/12 have different endpoints. Contrast this with 2/3 = 4/6, where both counts are multiplied by two.

h3. How could you rename 3/4 using twelfths?

Answer: Split each fourth into three twelfths. Three groups of three make 9/12.

Look for:

• Identify the factor from fourths to twelfths as three.
• Multiply the three selected fourths by three to obtain nine twelfths.
• State 3/4 = 9/12 and preserve the same whole or shaded endpoint.

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

h4. Can unequal-length strips prove equal physical amounts? Explain.

Answer: Equal fraction labels describe relative shares, but unequal wholes do not establish equal lengths.

Look for:

• Distinguish equal fraction values from equal physical lengths.
• Identify that the strips use unequal wholes.
• Explain that matching relative shares on those strips alone cannot prove equal lengths.

Teacher response: Assumes equal fraction labels guarantee equal physical amounts even when the wholes differ. Compare half of a 12 cm strip with half of a 6 cm strip. Measure 6 cm and 3 cm. Keep the fraction label visible and ask what changed; then repeat the comparison using two equal-length strips.

e1. Compare 3/6 and 1/2. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.

e2. Explain why 1/2 and 2/4 are equivalent.

Answer: Repartition the same whole: each half becomes two fourths, and the shaded endpoint stays fixed.

Look for:

• Keep the whole fixed when moving from halves to fourths.
• Split each half into two equal fourths, giving 1/2 = 2/4.
• Explain that the shaded amount or endpoint stays unchanged.

Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.