HOW IT WORKS
A class of 26 students is going on a trip. Each van holds 6 students. How many vans do they need?
Step 1 — Divide. 26 ÷ 6 = 4 remainder 2.
Step 2 — Say what the numbers mean. 4 vans get completely full. 2 students are still standing there.
Step 3 — Ask: can I leave the leftover behind? No. Those 2 students still have to get on a van.
Step 4 — Answer the question that was asked. They need 5 vans — 4 full ones and 1 with only 2 students in it.
The trap: the answer is not "4 remainder 2." Always go back and re-read what the question asked for. Sometimes you round up (vans, boxes, tables), sometimes you throw the leftover away (how many full boxes did we fill?).
1.GUIDED
Musa is packing 34 books into boxes. Each box holds 8 books. How many boxes does he need so that every book is packed?
Divide: 34 ÷ 8 = remainder
What does the remainder mean? books left over
Can those books be left behind?
So how many boxes?
Answer:
Did you know? Libraries pack books spine-up in boxes so the pages don't bend under the weight.
2.
A baker has 50 cupcakes. He puts 7 cupcakes in each gift box. How many completely full gift boxes can he make, and how many cupcakes are left over?
SHOW YOUR WORK
Answer:
Did you know? Cupcakes got their name because the early recipes measured everything in cups instead of weighing it.
3.
45 students are going to a museum. Each bus holds 9 students. How many buses are needed? Careful — read the question twice.
SHOW YOUR WORK
Answer:
Did you know? School buses are yellow because your eye notices yellow faster than any other color out of the corner of your vision.
4.
A camp has 38 kids and each tent sleeps 5. Every kid needs a place to sleep. How many tents does the camp need, and how many kids are in the last tent?
SHOW YOUR WORK
Answer:
Did you know? A tent stays up because the poles push out while the ropes pull in — the two forces balance each other.
HOW IT WORKS
Yusuf eats 1/2 of a pizza. Zaid eats 1/3 of the same pizza. How much did they eat together?
Step 1 — Notice the problem. Halves and thirds are different-sized pieces. You cannot add 1 + 1 and get an answer, because the pieces aren't the same size.
Step 2 — Find a size they can both become. Cut everything into sixths. Both 2 and 3 divide into 6.
Step 3 — Rewrite each fraction. 1/2 = 3/6 (multiply top and bottom by 3). 1/3 = 2/6 (multiply top and bottom by 2).
Step 4 — Now add. 3/6 + 2/6 = 5/6. The bottom number does not change — you're just counting sixths.
The trap: 1/2 + 1/3 is not 2/5. Never add the bottoms. To find the common bottom, the fastest way is to multiply the two denominators together (2 × 3 = 6) — it always works, even if it isn't always the smallest.
1.GUIDED
Yusuf reads 1/4 of his book on Monday and 1/3 of it on Tuesday. What fraction of the book has he read in all?
Common bottom number: 4 × 3 =
Rewrite: 1/4 = and 1/3 =
Add the tops, keep the bottom:
Answer:
Did you know? The line in a fraction is called a vinculum, and it has been used in math for over 800 years.
2.
A recipe needs 2/3 cup of flour and 1/6 cup of sugar. How many cups of dry ingredients is that altogether?
SHOW YOUR WORK
Answer:
Did you know? Bakers weigh flour instead of scooping it, because a packed cup can hold 20% more than a loose one.
3.
Bilal ran 3/4 of a mile and then walked 2/5 of a mile. How far did he go in total? Is it more or less than 1 mile?
SHOW YOUR WORK
Answer:
Did you know? A mile was originally 1,000 paces of a Roman soldier — mille passus means "a thousand steps."
4.
A water jug is 7/8 full. Yusuf pours out 1/3 of a jug. What fraction of the jug is left? (Same steps — but subtract this time.)
SHOW YOUR WORK
Answer:
Did you know? A gallon of water weighs about 8 pounds, which is why a full jug is so hard to lift with one hand.
3rd Grade (remainders)
1. 5 boxes.
34 ÷ 8 = 4 r 2. Four full boxes, 2 books left. Books can't be left behind → 5 boxes (last one has 2).
2. 7 full boxes, 1 cupcake left over.
50 ÷ 7 = 7 r 1. This one asks for completely full boxes, so we do NOT round up: 7 boxes, 1 cupcake left.
3. 5 buses.
45 ÷ 9 = 5 exactly, no remainder. The "read twice" warning is a fake-out — sometimes it just comes out even. 5 buses.
4. 8 tents; 3 kids in the last one.
38 ÷ 5 = 7 r 3. Every kid needs a bed → round up to 8 tents; the 8th holds the leftover 3.
4th Grade (unlike denominators)
1. 7/12 of the book.
Common bottom 4 × 3 = 12. 1/4 = 3/12, 1/3 = 4/12. 3/12 + 4/12 = 7/12.
2. 5/6 cup.
6 already works (3 divides into 6). 2/3 = 4/6. 4/6 + 1/6 = 5/6. Good one to point out that you don't always need to multiply the bottoms.
3. 23/20 miles = 1 3/20 miles — more than 1 mile.
Common bottom 4 × 5 = 20. 3/4 = 15/20, 2/5 = 8/20. 15/20 + 8/20 = 23/20, which is 1 whole and 3/20. Since 23 > 20, it's more than a mile.
4. 13/24 of the jug.
Common bottom 8 × 3 = 24. 7/8 = 21/24, 1/3 = 8/24. 21/24 − 8/24 = 13/24. Same procedure as adding — only the last step changes.