Common Denominator Kitchen: 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. Model one third cup plus one fourth cup using equal-sized cup diagrams.

activity-2. Repartition both into twelfths and count seven twelfths in all.

activity-3. Partners prepare a second recipe and estimate before adding.

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: Model one third cup plus one fourth cup using equal-sized cup diagrams.

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

phase-3. 10 min · partner: Repartition both into twelfths and count seven twelfths in all.

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

phase-4. 7 min · discussion: Partners prepare a second recipe and estimate before adding.

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 1/3 + 1/4. Show a model or your reasoning. Why must the pieces have the same size before adding?

Answer: Students: Complete the two exit questions independently.

d1. A learner writes 1/3 + 1/4 = 2/7. What is wrong with the units?

Answer: Thirds and fourths are different sizes; rename as 4/12 + 3/12 = 7/12.

Teacher response: If the learner combines numerator and denominator counts directly. Place a third piece beside a fourth piece and ask what equal-sized piece can measure both; partition the common whole into twelfths.

d2. Could 2/7 be the sum of the positive amounts 1/3 and 1/4?

Answer: No. 2/7 is less than 1/3, while adding a positive amount must increase 1/3.

Teacher response: If the learner accepts the computed answer without checking its size. Mark 1/3 on a line and show that a positive addition moves right; compare 2/7 = 6/21 with 1/3 = 7/21.

r1. Find 1/2 + 1/3 and state why the denominator is not five.

Answer: 3/6 + 2/6 = 5/6. The two amounts are counted in sixths, not fifths.

e1. Calculate 1/3 + 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

e2. Why must the pieces have the same size before adding?

Answer: A numerator counts units; both counts must refer to the same fractional unit.

Look for:

• Explain that each numerator is a count of a named fractional unit.
• Identify the different unit sizes in 1/3 and 1/4.
• Say that equivalent common units allow those counts to be combined without changing either amount.

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

teach-1. Identify the size of each addend’s unit and choose a common partition.

teach-2. Scale both counts in each fraction to preserve the original amount.

teach-3. Combine only the common-unit numerators and compare the result with the two positive addends.

next-1. e1 meets its criterion: e1 gives 7/12 for 1/3 + 1/4. e2 also meets its criterion: e2 explains that numerator counts can be combined only when they name the same fractional unit.

Teacher response: Continue to the linked next lesson. Use the same common-unit structure to remove an amount and check by addition. Next original lesson: /lessons/fraction-difference-trail/.

next-2. e1 does not yet meet this criterion: e1 gives 7/12 for 1/3 + 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 explains that numerator counts can be combined only when they name the same fractional unit.

Teacher response: Ask the learner to point to one third and one fourth, then to the twelfth-piece used to count both. Require that unit explanation in e2. Short intervention: /interventions/unlike-estimation/.

c1. Calculate 1/3 + 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c2. Calculate 2/5 + 1/2. Show a model or your reasoning.

Answer: 9/10

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c3. Calculate 3/4 + 1/6. Show a model or your reasoning.

Answer: 11/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c4. Calculate 1/2 + 2/3. Show a model or your reasoning.

Answer: 7/6

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c5. Calculate 5/8 + 1/4. Show a model or your reasoning.

Answer: 7/8

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c6. Calculate 2/3 + 3/5. Show a model or your reasoning.

Answer: 19/15

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c7. Calculate 1/6 + 3/8. Show a model or your reasoning.

Answer: 13/24

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c8. Calculate 3/10 + 2/5. Show a model or your reasoning.

Answer: 7/10

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

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

Answer: 3/4

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

s2. Calculate 1/3 + 1/6. Show a model or your reasoning.

Answer: 1/2

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

s3. Calculate 1/4 + 1/8. Show a model or your reasoning.

Answer: 3/8

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

s4. Calculate 1/5 + 2/5. Show a model or your reasoning.

Answer: 3/5

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. Explain 1/3 + 1/4 with a model.

Answer: Four twelfths plus three twelfths gives seven twelfths.

Look for:

• Use one fixed whole partitioned into twelfths.
• Represent 1/3 with four twelfths and 1/4 with three twelfths.
• Combine the selected parts to show 7/12.

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

h2. Why is 1/3 + 1/4 not 2/7?

Answer: The pieces have different sizes; the denominators cannot simply be added.

Look for:

• Identify thirds and fourths as different-sized pieces.
• Explain why adding the two denominators does not create a valid common unit.
• Show a correction, such as 4/12 + 3/12 = 7/12.

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

h3. Find two common denominators for thirds and fourths.

Answer: 12 and 24 both work and produce equivalent sums.

Look for:

• Give two different common denominators divisible by both three and four.
• Use them to rename thirds and fourths equivalently, such as 4/12 and 3/12 or 8/24 and 6/24.
• Explain that both common-unit sums have the same value.

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. Invent a recipe using unlike fractions totaling one cup.

Answer: Use 1/2 cup of oats, 1/4 cup of seeds and 1/4 cup of dried fruit: the total is one cup.

Look for:

• Create a recipe with at least two unlike fractional amounts in cups.
• Make the quantities non-overlapping ingredient amounts that are combined.
• Verify the exact total is one cup using equivalent fractions.

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. Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

e1. Calculate 1/3 + 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

e2. Why must the pieces have the same size before adding?

Answer: A numerator counts units; both counts must refer to the same fractional unit.

Look for:

• Explain that each numerator is a count of a named fractional unit.
• Identify the different unit sizes in 1/3 and 1/4.
• Say that equivalent common units allow those counts to be combined without changing either amount.

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.