Division with remainders: a lesson in what is left over
The fourth division lesson, for the day the candies do not share out. Fourteen counters are drawn already dealt into four rings with two left over outside them, the sum is worked step by step to 3 r 2, and the check is written the way it should be — 4 × 3 + 2 = 14 — before five more to try with the same picture, and one story about tables.
Read the box and count what is left outside the rings. Then try the six below.
Sometimes it doesn’t share out evenly
14 ÷ 4 means 14 candies shared between 4 children. Deal them out and 2 are left in the bag — not enough for everyone to get one more. What is left over is the remainder.
2.Only 2 left — not enough for everyone to get one more. Stop.
3.3 each, 2 left over. 14 ÷ 4 = 3 remainder 2.
4.Check: 4 × 3 = 12, and 12 + 2 = 14.
14 ÷ 4 =3 r 2
4 × 3 + 2 =14
When the leftover changes the answer
7 children. Each car holds 3. 7 ÷ 3 = 2 r 1. Two cars are full and one child is still waiting — so you need 3 cars. The remainder made the answer go up.
For the grown-up: when they can’t give everyone one more, stop — what’s in the bag is the remainder. Then ask what the leftover means in the story.
Division with remainders
NameDate/ 6
Read the box and count what is left outside the rings. Then try the six below.
1.23 ÷ 4 =
2.17 ÷ 5 =
3.9 children sit 4 to a table. How many tables?
4.11 ÷ 3 =
5.13 ÷ 4 =
6.20 ÷ 6 =
Division with remainders
NameDate/ 6
Read the box and count what is left outside the rings. Then try the six below.
1.23 ÷ 4 =5 r 3
2.17 ÷ 5 =3 r 2
3.9 children sit 4 to a table. How many tables?3
4.11 ÷ 3 =3 r 2
5.13 ÷ 4 =3 r 1
6.20 ÷ 6 =3 r 2
What is on this sheet
A remainder is drawn before it is written. The two counters that would not go round sit outside every ring, which is what a leftover looks like on a table, and the answer is written as 3 r 2 with the check as a multiplication sentence rather than a chain of equals signs — because 3 r 2 is not a number, and a child who writes 14 ÷ 4 = 3 r 2 = 4 × 3 + 2 has been taught that it is. The five counter problems keep every total under two dozen and every divisor at what a child can ring.
The sixth problem is a story, and it is the reason this page exists: nine children sit four to a table, and the answer is three tables, not 2 r 1. Children who can do the division perfectly still stop at the remainder when the story wants it rounded up — it is the best-known failure in the whole research on division — so the lesson works one such story on the page before asking for one. The problems run on to a second page; a parent who wants the lesson alone prints page one.
The answers were worked out when the problems were, so the key is the same page with the answers drawn in rather than a second attempt at the same questions — there is no way for the two to disagree. The number in the footer is the seed the sheet was built from: it is printed so that the identical sheet can be had again next week, rather than one that merely looks like it.
Change what is on it
This page is one sheet out of a family that can print thousands, and the settings behind it — how many problems, how big the numbers get, how they are laid out, whether there is room to work — are all switches. The builder link at the top of the page carries this sheet’s own settings, so the bench opens on exactly the sheet here rather than on a blank one.
The configuration travels in the address bar rather than on a server, so the link can be bookmarked or sent to somebody as it stands — and what they open is this sheet, down to the seed, rather than another one like it.
The same facts, on screen
The Grid’s division deck has no remainders in it, on purpose: once a child has this page, the exact facts are the ones to race.
