Scaling Stories
Predict how a multiplier below, equal to or above one changes a quantity.
Preview the complete lesson
Eight core questions and a two-question exit ticket. Print one set per student.
c1. Calculate 3/4 × 1/2. Show a model or your reasoning.
c2. Calculate 3/4 × 1. Show a model or your reasoning.
c3. Calculate 3/4 × 2. Show a model or your reasoning.
c4. Calculate 2/3 × 3/4. Show a model or your reasoning.
c5. Calculate 2/3 × 3/2. Show a model or your reasoning.
c6. Calculate 5/6 × 2/5. Show a model or your reasoning.
c7. Calculate 4/5 × 5/4. Show a model or your reasoning.
c8. Calculate 7/8 × 4/3. Show a model or your reasoning.
e1. Calculate 3/4 × 1/2. Show a model or your reasoning.
e2. State the scaling rule for positive amounts.
Before the lesson
Recognize a whole and count equal parts.
Materials
- Pencil: 1 per student
- Plain paper sheets: 2 per student
- Projector: 1 per class (optional)
Activity
- Start with a 3/4 meter ribbon. Predict the result of scaling by 1/2, 1 and 2.
- Model each product and compare it with the original length.
- Partners write a rule based on the multiplier, then test a new ribbon length.
45-minute teaching sequence
Look at the first model or task. What is the whole? What do you notice?
Start with a 3/4 meter ribbon. Predict the result of scaling by 1/2, 1 and 2.
Model each product and compare it with the original length.
Partners write a rule based on the multiplier, then test a new ribbon length.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Calculate 3/4 × 1/2. Show a model or your reasoning. State the scaling rule for positive amounts.
Watch for this misconception
Assuming multiplication always increases value.
Locate the multiplier relative to one before predicting the product.
Teacher answer key. Open responses include a rubric and one valid sample.
c1. Calculate 3/4 × 1/2. Show a model or your reasoning.
3/4 × 1/2 = 3/8. Multiply the numerators and denominators; interpret the groups or overlap.
c2. Calculate 3/4 × 1. Show a model or your reasoning.
3/4 × 1 = 3/4. Multiply the numerators and denominators; interpret the groups or overlap.
c3. Calculate 3/4 × 2. Show a model or your reasoning.
3/4 × 2 = 3/2. Multiply the numerators and denominators; interpret the groups or overlap.
c4. Calculate 2/3 × 3/4. Show a model or your reasoning.
2/3 × 3/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
c5. Calculate 2/3 × 3/2. Show a model or your reasoning.
2/3 × 3/2 = 1. Multiply the numerators and denominators; interpret the groups or overlap.
c6. Calculate 5/6 × 2/5. Show a model or your reasoning.
5/6 × 2/5 = 1/3. Multiply the numerators and denominators; interpret the groups or overlap.
c7. Calculate 4/5 × 5/4. Show a model or your reasoning.
4/5 × 5/4 = 1. Multiply the numerators and denominators; interpret the groups or overlap.
c8. Calculate 7/8 × 4/3. Show a model or your reasoning.
7/8 × 4/3 = 7/6. Multiply the numerators and denominators; interpret the groups or overlap.
s1. Calculate 1/2 × 1/2. Show a model or your reasoning.
1/2 × 1/2 = 1/4. Multiply the numerators and denominators; interpret the groups or overlap.
s2. Calculate 1/2 × 1. Show a model or your reasoning.
1/2 × 1 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
s3. Calculate 1/2 × 2. Show a model or your reasoning.
1/2 × 2 = 1. Multiply the numerators and denominators; interpret the groups or overlap.
s4. Calculate 3/4 × 1/2. Show a model or your reasoning.
3/4 × 1/2 = 3/8. Multiply the numerators and denominators; interpret the groups or overlap.
h1. Why is 3/4 × 1/2 smaller than 3/4?
Taking half of a positive amount makes it smaller.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
h2. Give a counterexample to multiplication always makes numbers bigger.
1/2 × 1/2 = 1/4, smaller than 1/2.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
h3. What happens when a positive quantity is scaled by one?
Its value stays unchanged.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
h4. Predict without computing whether 7/8 × 4/3 exceeds 7/8.
It exceeds it because the multiplier 4/3 is above one.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
e1. Calculate 3/4 × 1/2. Show a model or your reasoning.
3/4 × 1/2 = 3/8. Multiply the numerators and denominators; interpret the groups or overlap.
e2. State the scaling rule for positive amounts.
A factor below one shrinks, one preserves, and above one enlarges the amount.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
s1. Calculate 1/2 × 1/2. Show a model or your reasoning.
s2. Calculate 1/2 × 1. Show a model or your reasoning.
s3. Calculate 1/2 × 2. Show a model or your reasoning.
s4. Calculate 3/4 × 1/2. Show a model or your reasoning.
e1. Calculate 3/4 × 1/2. Show a model or your reasoning.
e2. State the scaling rule for positive amounts.
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
h1. Why is 3/4 × 1/2 smaller than 3/4?
h2. Give a counterexample to multiplication always makes numbers bigger.
h3. What happens when a positive quantity is scaled by one?
h4. Predict without computing whether 7/8 × 4/3 exceeds 7/8.
e1. Calculate 3/4 × 1/2. Show a model or your reasoning.
e2. State the scaling rule for positive amounts.
Teaching notes & lesson sources
Designed for Grade 5. Predict how a multiplier below, equal to or above one changes a quantity.
5.NF.B.5b curriculum reference
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.