Solve rate, total-amount and number-of-units problems

What you will learn

• Solve rate, total-amount and number-of-units problems

Models used: number sentences, stories, tables.

Before you start

• Interpret amount per unit

Find the price of one

Three notebooks cost $12. Each notebook costs the same.

1. All the small boxes represent equally sized units, even when they belong to different rows.

   Three notebooks cost $12. Each notebook costs the same.

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Notebook cost ($)

   3 equal units · 3 units = 12

2. All the small boxes represent equally sized units, even when they belong to different rows.

   Share $12 equally between the three notebooks. Each share is $4.

   $12 ÷ 3 = $4

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Notebook cost ($)44412 altogether

   3 equal units · 3 units = 12 · 1 unit = 4

3. All the small boxes represent equally sized units, even when they belong to different rows.

   Now change only the number of notebooks: buy five at the same $4 each. Five equal shares give the new total.

   5 × $4 = $20

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Notebook cost ($)4444420 altogether

   5 equal units · 5 units = 20 · 1 unit = 4

Worked example: A new example, together

One notebook costs $6. What do 4 notebooks cost?

1. Repeat the same price for each notebook. Read each value with the heading above its column.

   Read both unit labels and identify the rate, total amount and number of units.

   Each unit contributes the same amount, so equal-group reasoning applies.

   Repeat the same price for each notebook.

   Cost per notebook ($)	Notebooks
   6	4

2. Repeat the same price for each notebook. Read each value with the heading above its column.

   One notebook costs $6. We need that same amount 4 times.

   Repeat the same price for each notebook.

   Cost per notebook ($)	Notebooks
   6	4

   One notebook costs $6. What do 4 notebooks cost?

   24 dollars

3. Repeat the same price for each notebook. Read each value with the heading above its column.

   Join 4 equal costs; do not add the number of books to their price.

   Repeat the same price for each notebook.

   Cost per notebook ($)	Notebooks
   6	4

4. Repeat the same price for each notebook. Read each value with the heading above its column.

   4 × $6 = $24.

   Repeat the same price for each notebook.

   Cost per notebook ($)	Notebooks
   6	4

5. Repeat the same price for each notebook. Read each value with the heading above its column.

   Check the result against the question we started with.

   4 × $6 = $24.

   We found 24 dollars. Each unit contributes the same amount, so equal-group reasoning applies.

   Repeat the same price for each notebook.

   Cost per notebook ($)	Notebooks
   6	4

Common mistakes to discuss

• Using the total as the rate. Decide which of the three is missing. Total = rate × units, so 5 notebooks at $3 cost $15, and $15 for 5 gives $3 each.

Check your understanding

• Solve rate, total-amount and number-of-units problems
• connect a rate with its total and number of units
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.