Fraction Difference Trail: 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 a trail of length 5/6 mile and mark the first 1/4 mile as completed.

activity-2. Use twelfths to find the remaining trail.

activity-3. Partners estimate a difference, compute it, then check it by adding back.

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 a trail of length 5/6 mile and mark the first 1/4 mile as completed.

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

phase-3. 10 min · partner: Use twelfths to find the remaining trail.

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

phase-4. 7 min · discussion: Partners estimate a difference, compute it, then check it by adding back.

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: Calculate 5/6 - 1/4. Show a model or your reasoning. What addition equation checks a subtraction?

Answer: Students: Complete the two exit questions independently.

d1. Rename both quantities before finding 5/6 − 1/4.

Answer: 10/12 − 3/12 = 7/12.

Teacher response: If the learner subtracts 5 − 1 and 6 − 4 to obtain 4/2. Build ten twelfths and physically remove three; keep the size label on every remaining piece.

d2. How would you check a proposed difference of 7/12 for 5/6 − 1/4?

Answer: Add back 1/4: 7/12 + 3/12 = 10/12 = 5/6.

Teacher response: If the learner adds the difference to the starting amount instead of the removed amount. Label three roles on a bar: starting amount, removed portion, and remainder. Reassemble the removed portion with the remainder.

r1. Calculate 3/4 − 1/3 and check it by addition.

Answer: 5/12; 5/12 + 4/12 = 9/12 = 3/4.

e1. Calculate 5/6 - 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

e2. What addition equation checks a subtraction?

Answer: Difference plus the removed amount must equal the starting amount.

Look for:

• State difference + removed amount = starting amount.
• Identify those roles in e1.
• Give or explain 7/12 + 1/4 = 5/6 as the check.

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

teach-1. Show the starting fraction in a common unit with the removed amount.

teach-2. Cross out only the number of common-unit pieces being removed.

teach-3. Reattach those pieces to the remainder and write the matching addition check.

next-1. e1 meets its criterion: e1 gives 7/12 for 5/6 − 1/4. e2 also meets its criterion: e2 states difference + removed amount = starting amount, or applies 7/12 + 1/4 = 5/6.

Teacher response: Continue to the linked next lesson. Add an exchange of one whole when the fractional part is too small to remove the requested amount. Next original lesson: /lessons/mixed-number-repair/.

next-2. e1 does not yet meet this criterion: e1 gives 7/12 for 5/6 − 1/4.

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/unlike-estimation/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 states difference + removed amount = starting amount, or applies 7/12 + 1/4 = 5/6.

Teacher response: Ask the learner to label the removed 1/4 and the remaining 7/12 on e1, then join them to construct the e2 check. Short intervention: /interventions/unlike-estimation/.

c1. Calculate 5/6 - 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c2. Calculate 3/4 - 1/3. Show a model or your reasoning.

Answer: 5/12

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c3. Calculate 7/8 - 1/2. Show a model or your reasoning.

Answer: 3/8

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c4. Calculate 4/5 - 1/6. Show a model or your reasoning.

Answer: 19/30

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c5. Calculate 2/3 - 1/5. Show a model or your reasoning.

Answer: 7/15

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c6. Calculate 9/10 - 3/4. Show a model or your reasoning.

Answer: 3/20

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c7. Calculate 5/8 - 1/3. Show a model or your reasoning.

Answer: 7/24

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c8. Calculate 7/9 - 1/6. Show a model or your reasoning.

Answer: 11/18

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

s1. Calculate 3/4 - 1/2. Show a model or your reasoning.

Answer: 1/4

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

s2. Calculate 2/3 - 1/3. Show a model or your reasoning.

Answer: 1/3

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.

s3. Calculate 5/6 - 1/2. Show a model or your reasoning.

Answer: 1/3

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

s4. Calculate 3/5 - 1/5. Show a model or your reasoning.

Answer: 2/5

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.

h1. Estimate 5/6 - 1/4 before calculating.

Answer: It is a little more than one half; exact result 7/12.

Look for:

• Give a reasonable estimate near and above one half for 5/6 − 1/4.
• Support the direction or bound, for example 5/6 > 3/4 implies the difference exceeds 1/2.
• Check that the exact result 7/12 fits the estimate.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound.

h2. Check 3/4 - 1/3 by adding back.

Answer: 5/12 + 1/3 = 9/12 = 3/4.

Look for:

• Identify the difference 5/12.
• Write 5/12 + 1/3 to restore the removed amount.
• Rename 1/3 as 4/12 and recover 9/12 = 3/4.

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

h3. Why cannot denominators be subtracted?

Answer: They describe unit sizes, which must be matched, not removed.

Look for:

• Identify denominators as naming fractional unit sizes.
• Explain that subtraction removes counts of matched-size pieces.
• Use a common-unit example to show that subtracting denominators changes the unit incorrectly.

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

h4. Write a trail story with a remainder of 1/2 mile.

Answer: A 3/4 mile trail with 1/4 mile completed leaves 1/2 mile.

Look for:

• Write a trail situation with a stated starting length and completed length.
• Subtract the completed length from the starting length.
• Verify a remainder of exactly 1/2 mile with consistent units.

Teacher response: Chooses an operation from the numbers without identifying whether the task combines, takes a part, shares, or counts groups. Ask the learner to finish “I know …; I need …” with units. Act out the quantity: combine disjoint colors, overlap two fractional selections, share one amount, or count fixed-size pieces. Write an equation only after the action is identified.

e1. Calculate 5/6 - 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

e2. What addition equation checks a subtraction?

Answer: Difference plus the removed amount must equal the starting amount.

Look for:

• State difference + removed amount = starting amount.
• Identify those roles in e1.
• Give or explain 7/12 + 1/4 = 5/6 as the check.

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.