The Lumenary · Grades 3–5

Whole-number arithmetic

Name: ______________________

Date: ______________________

You will learn to

Explain multiplication as equal groups and as arrays, and use an array to see why $a \times b = b \times a$
Use the commutative, associative and distributive laws to calculate flexibly, for example $7 \times 98 = 7 \times 100 - 7 \times 2$
Multiply multi-digit numbers and explain every step of the written method through place value and the distributive law
Divide with remainder, writing $a = q \times b + r$ with $0 \le r < b$ , and explain why $q$ and $r$ are unique
Explain why dividing by zero has no answer

1.
Use the distributive law to work out $8 \times 99$ in your head. (Hint: $99 = 100 - 1$ .)
2.
Amina works out $4 \times 17 \times 25$ like this: $4 \times 17 \times 25 = 17 \times (4 \times 25) = 17 \times 100 = 1700$ . Which laws make her method correct?
- A.The commutative and associative laws
- B.Only the distributive law
- C.No law: she changed the problem, so her answer is only a guess
- D.The rule that multiplying makes numbers bigger
3.
Use the area model to work out $36 \times 24$ . Split $36 = 30 + 6$ and $24 = 20 + 4$ , and add all four pieces.
4.
Divide $100$ by $7$ . What is the remainder?
5.
Kofi says $47 \div 8 = 4$ remainder $15$ . He checks: $4 \times 8 + 15 = 32 + 15 = 47$ . Is his answer right?
- A.No. His check is true, but $15$ is not less than $8$ . The answer is $5$ remainder $7$ .
- B.Yes, because his check gives $47$ .
- C.No, because $4 \times 8 + 15$ is not $47$ .
- D.No. The answer is $5$ remainder $8$ .
6.
A farm has $1000$ eggs, and each tray holds $30$ eggs. How many trays are needed to hold all the eggs?
7.
What is $6 \div 0$ ?
- A. $0$
- B. $6$
- C.There is no answer: dividing by zero is undefined
- D.Infinity
8.
Which of these products are not bigger than the first number? (Choose all that apply.)
- A. $9 \times 1$
- B. $9 \times 0$
- C. $9 \times 2$
- D. $1 \times 9$
9.
In Lean, a / b is the quotient and a % b is the remainder when a is divided by b. Gideon wants to state precisely that the remainder is smaller than the number we divide by (for b > 0). Which statement says exactly that?
- A.a % b < b
- B.a % b ≤ b
- C.a / b < b
- D.a < b
10.
Write a function divide(a, b) that finds the quotient and remainder by repeated subtraction: keep taking b away from a while at least b is left, counting how many times. Return the pair (q, r). You may assume a >= 0 and b > 0.
Computational — Practice

Answer key — Whole-number arithmetic

1.
$8 \times 99 = 8 \times 100 - 8 \times 1 = 800 - 8 = 792$ . The distributive law lets us split $99$ into $100 - 1$ .
2.
She used the commutative law to swap $4$ and $17$ , and the associative law to multiply $4 \times 25 = 100$ first: $4 \times 17 \times 25 = 17 \times 4 \times 25 = 17 \times (4 \times 25) = 17 \times 100 = 1700$ .
3.
$36 \times 24 = 30 \times 20 + 30 \times 4 + 6 \times 20 + 6 \times 4 = 600 + 120 + 120 + 24 = 864$ .
4.
$14 \times 7 = 98$ and $15 \times 7 = 105$ is too big. So $100 = 14 \times 7 + 2$ , and since $2 < 7$ , the remainder is $2$ .
5.
Division with remainder needs $a = q \times b + r$ and $0 \le r < b$ . Kofi's equation is true, but $15 \ge 8$ . Taking one more $8$ from the $15$ gives $47 = 5 \times 8 + 7$ , with $7 < 8$ . So $47 \div 8 = 5$ remainder $7$ .
6.
$1000 = 33 \times 30 + 10$ . So $33$ trays are filled, and the last $10$ eggs need one more tray: $34$ trays in all. The remainder tells us whether to round the quotient up.
7.
If $6 \div 0 = q$ , then $q \times 0 = 6$ . But $q \times 0 = 0$ for every $q$ , so no $q$ works. Division by zero is undefined.
8.
$9 \times 1 = 9$ and $9 \times 0 = 0$ are not bigger than $9$ . But $9 \times 2 = 18 > 9$ and $1 \times 9 = 9 > 1$ . So multiplying does not always make bigger: multiplying by $1$ changes nothing, and multiplying by $0$ gives $0$ .
9.
a % b < b. The remainder is a % b, and it must be strictly less than b. In the lesson's Lean file this is the second half of div_mod_library, which uses the library theorem Nat.mod_lt.
10.
```
def divide(a, b):
    q = 0
    while a >= b:
        a = a - b
        q = q + 1
    return (q, a)
```
The loop stops as soon as fewer than b are left, so the remainder is always less than b. The tests compare your function with Python's built-in divmod on many cases. That is evidence for those cases. The proof in the lesson shows the method works for every a and every b > 0.