Before the bell
- The I Do form in the I Do deck. All three sections. If you are here, this should be done.
- The We Do: Divide and Remainder: Whole Numbers Only. Read each page with the class. Work every Try It in your notebook, then check it. Nothing to turn in.
- Bench Basics: Divide and Remainder. About 30 minutes. It opens later in class on the Bench Basics slide. Write your credit code in your notebook.
- The exit ticket. The last 10 minutes of class, a locked form on a school Chromebook. It asks the same kinds of questions as the Try Its and the bench.
Divide and Remainder: Whole Numbers Only
This section shows the three kinds of steps on today's bench: dividing whole numbers, working a line in the right order, and following a box as its value changes.
The Whole Part and the Remainder
When a computer divides two whole numbers, it often keeps only the whole-number part of the answer and drops the rest. The part that is dropped can be worked out separately, and it is called the remainder. This section writes whole-number division as ÷ and the remainder as rem. It writes * for times, which is the symbol Java and the bench use for multiplication. In November, the class will see Java write all three as shown below.
1whole part: ÷ in this section, / in Java 2left over: rem in this section, % in Java 3times: * in this section, * in Java
How to find a ÷ b and a rem b:
- Find the largest number of whole times b fits into a. That number is a ÷ b.
- Multiply it by b, and subtract the product from a. What is left is a rem b.
- Check the work: b times the whole part, plus the remainder, gives back a.
Example 1. Find 17 ÷ 5 and 17 rem 5.
15 * 3 = 15 fits 25 * 4 = 20 too big, so 17 ÷ 5 is 3 317 - 15 = 2 so 17 rem 5 is 2 4check: 5 * 3 + 2 = 17
The answer is 3, not 3.4, because whole-number division keeps only the whole part.
Example 2. Find 4 ÷ 9 and 4 rem 9.
19 * 0 = 0 fits 29 * 1 = 9 too big, so 4 ÷ 9 is 0 34 - 0 = 4 so 4 rem 9 is 4 4check: 9 * 0 + 4 = 4
When a number is smaller than the divisor, the whole part is 0 and the remainder is the number itself.
Example 3. Find 36 ÷ 6 and 36 rem 6.
16 * 6 = 36 fits exactly, so 36 ÷ 6 is 6 236 - 36 = 0 so 36 rem 6 is 0
When a number divides evenly, the remainder is 0. The remainder is always smaller than the divisor, so n rem 6 can be 0, 1, 2, 3, 4 or 5, but never 6.
Work each one in your notebook first, then check it.
Try It 1. In whole-number division, what is 26 ÷ 8?
Try It 2. What is 26 rem 8?
Try It 3. What are 5 ÷ 12 and 5 rem 12?
Try It 4. Work 47 ÷ 6 the way Example 1 does. Give the whole part, the remainder and the check.
Check your answer
6 7 is 42, which fits, and 6 8 is 48, which is too big, so 47 ÷ 6 is 7. Then 47 - 42 is 5, so 47 rem 6 is 5. The check is 6 * 7 + 5 = 47.
One Step at a Time
A line with several operations is worked in a fixed order. First, work anything inside parentheses. Next, work *, ÷ and rem from left to right. After that, work + and - from left to right. Each step rewrites the whole line, so every new line shows one more operation finished.
Example 4. Work 20 - 14 ÷ 4.
120 - 14 ÷ 4 2= 20 - 3 ÷ comes before - 3= 17
Example 5. Work 7 + 8 ÷ 2 and (7 + 8) ÷ 2.
17 + 8 ÷ 2 (7 + 8) ÷ 2 2= 7 + 4 = 15 ÷ 2 parentheses first 3= 11 = 7
The two lines use the same numbers, but the parentheses change the order, and the order changes the answer.
Example 6. Work 7 ÷ 2 * 2.
17 ÷ 2 * 2 2= 3 * 2 ÷ and * go left to right 3= 6
The answer is 6, not 7, because 7 ÷ 2 drops the half before the times step.
Work each one in your notebook first, then check it.
Try It 5. Work this line one step at a time. What is its value?
130 - 12 ÷ 4
Try It 6. Work this line one step at a time. What is its value?
1(9 + 23) rem 4
Try It 7. Work this line one step at a time. What is its value?
19 ÷ 4 * 4
Boxes That Change
On the bench, a box holds one value, and an arrow puts a new value in it. A line that uses a box reads the value the box holds at that moment. A trace table records every box after each line runs.
Example 7. Find how many hours and minutes 135 minutes make.
1mins ← 135 2hours ← mins ÷ 60 3mins ← mins rem 60 4show hours 5show mins
line mins hours shown 1 135 2 135 2 3 15 2 4 15 2 2 5 15 2 15
Line 2 reads 135, because mins still holds 135 there. Line 3 replaces 135 with 15, so lines 4 and 5 show 2 and then 15.
Example 8. Find the last two digits of 472.
1n ← 472 2ones ← n rem 10 3n ← n ÷ 10 4tens ← n rem 10
line n ones tens 1 472 2 472 2 3 47 2 4 47 2 7
The remainder after dividing by 10 is the last digit, and ÷ 10 drops that digit. After line 3, n holds 47, so line 4 finds 7.
The bench catches three common mistakes, and each one changes the answer. The first is writing a decimal answer, such as 3.5 for 7 ÷ 2. The second is doing + or - before ÷ or rem. The third is using a box's old value after a line has replaced it.
Sources: Gosling et al., The Java Language Specification, Java SE 21 Edition (2023); Oracle, The Java Tutorials: Operators and Expressions, Statements, and Blocks; course repository, Bench Basics B08 job data (2026).
Work each one in your notebook first, then check it.
Try It 8. After line 3, what does secs hold?
1secs ← 200 2m ← secs ÷ 60 3secs ← secs rem 60
Try It 9. Trace these lines the way Example 8 does. What do n and ones hold after line 3?
1n ← 385 2ones ← n rem 10 3n ← n ÷ 10
Check your answer
Line 2 finds 385 rem 10, which is 5, so ones holds 5. Line 3 replaces n with 385 ÷ 10, which is 38. After line 3, n holds 38 and ones holds 5.
Divide and Remainder
Open the bench and work the jobs in order. They get harder as you go. Write your credit code in your notebook, then paste it into the credit-code form.
Bench Basics opens during your class period.
Open it on a school Chromebook. Locked mode does not work on a lab computer.
Open the exit ticketThe exit ticket opens 10 minutes before the bell.