Revisiting Remainders

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Purpose

This is a level 4 number link activity from the Figure It Out series. It relates to Stage 7 of the Number Framework. 

A PDF of the student activity is included.

Achievement Objectives
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
Student Activity

Click on the image to enlarge it. Click again to close. Download PDF (245 KB)

Specific Learning Outcomes

write a remainder as a fraction and as a decimal

Required Resource Materials

A calculator

FIO, Link, Number, Book Five, Revisiting Remainders, page 1

A classmate

Activity

Question 1 shows how the quotient or result in a division operation may be expressed in three different ways:
• as a whole number (with the remainder being left as a remainder: 10 r 13)
• as a mixed number (that is, as a whole number and fraction: 10 13/25)
• as a decimal number (10.52).
 

The method used depends on the problem or context. For example, if there are 263 people (students,teachers, and adult helpers) going on a trip and each bus has 25 seats, then it makes sense to think in terms of a remainder and order 11 buses. On the other hand, if the 263 were $263 from a restaurant bill to be shared among 25 people, then each person would pay $10.52. In practical terms, it would be unusual today to have a quotient such as 10 13/25 because most of our measurements are metric. But quotients involving mixed numbers could be used when there are common fractions involved, such as halves, quarters, and thirds. For example, if five apples are being shared evenly among four children, then each child will receive 1 1/4 apples (5 ÷ 4 = 1 1/4).
Question 1 deals with decimal amounts that are equivalent to fraction amounts. The students can convert an ordinary fraction into its decimal equivalent by dividing the numerator by the denominator, which can be done easily on a calculator. If the students have not already done so, they could find the decimal equivalents
of common fractions such as 1/4 , 1/2 , 3/4 , 1/3 , and 2/3 on a calculator. Of course, this problem also requires the students to understand that a fraction such as 3/4 is an alternative way of writing 3 ÷ 4 or 4√ 3.
As equivalent fractions, 1/4 = 25/100  = 0.25 and 1/2 = 5/10 = 0.5
Matiu’s table in question 3 shows how useful patterns are when working with number. The pattern involved in questions 3a and 3b is multiples of 15 plus 2, so the next four numbers are (3 x 15) + 2, (4 x 15) + 2, (5 x 15) + 2, and (6 x 15) + 2. When the students understand this, they can find any number in the series. For example, the tenth number would be (10 x 15) + 2, that is, 152.
Similarly, with question 3c, the students will quickly see that when they divide by 5, the quotient always ends in 0.4, and when they divide by 3, the quotient always ends in 0.666.  Ask them to consider why this happens. Some students will realise that it is because 2 (the remainder) ÷ 5 is 0.4 and 2 ÷ 3 is 0.666.
This is an opportunity to explain to students the symbol used to indicate a recurrent decimal (namely the dot over the 6 in 5.6).

Answers to Activity

1. a. Kirsty’s fraction 13/25 is correct, but it is not actually a remainder. (263 divides into 25 lots of 10 13/25 .)
Matiu is right, although he could have explained that 13 is the remainder. (There are 10 lots of 25 in 263 with 13 left over.)
Mei Ling is also right. (263 divides into 25 lots of 10.52.)
b. Answers may vary, but essentially 13/25 is equivalent to 52/100 or 0.52.
2. a. 5 2/10 (or 5 1/5) or 5.2 or 5 r 2
b. 9 2/4 (or 9 1/2) or 9.5 or 9 r 2
c. 7 3/6 (or 7 1/2) or 7.5 or 7 r 3
d. 4 36/50 (or 4 18/25) or 4.72 or 4 r 36
e. 6 5/8 or 6.625 or 6 r 5
f. 9 13/20 or 9.65 or 9 r 13
g. 19 12/25 or 19.48 or 19 r 12
h. 8 3/9 (8 1/3) or 8.333 or 8 r 3
i. 568/9 or 56.888 or 56 r 8

3a.

answer.
b. Answers are likely to vary. All the numbers that work end in 7 or 2. After the first number, 17, all solutions are obtained by adding 15 (3 x 5) to the previous solution. Another way of describing the pattern is adding 2 to each multiple of 15 (15 + 2 = 17, 30 + 2 = 32, and so on).

c.

table.
Strategies could include:
For dividing by 3, consider the nearest multiple of 3 that is lower than the “number that works” and add 0.666 to the other factor that makes that multiple.
(For example: 3 x 5 = 15. 17 ÷ 3 = 5.666)
For dividing by 5, go to the nearest multiple of 5 that is lower than the “number that works” and add 0.4 to the other factor that makes that multiple.
(For example: 5 x 3 = 15. 17 ÷ 5 = 3.4)

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Level Four