Solve problems that involve adding and subtracting decimals.

Number Framework Stage 7

Calculators

### Background Maths

A major misconception for the students is to treat decimals as two independent sets of

whole numbers separated by a decimal point.

Students who believe that 0.8 is less than 0.75 because 75 is greater than

8 show signs of this misconception. Providing operational problems where the

students must work across the decimal point helps to address the problem. Suggest to

students that the decimal point is a marker for the ones place rather than a “barrier”

that separates the wholes from the fractions.

This link between the digits on one side of the decimal point and the other is further

strengthened when students encounter decimals in the context of division where there

is a remainder.

### Using Materials

Make 10 candy bars by joining 10 Unifix cubes to form each bar. Wrap each bar in a

paper sleeve to establish its “oneness”. Tell the students that this brand of candy only

comes in bars of 10 cubes.

Pose the problem: “If there are eight bars of candy and four friends, how much candy

will each person get?” (two whole bars)

Record the result as: 8 ÷ 4 = 2 wholes. Note that a whole means one.

Pose the problem: “If there were six candy bars and five friends, how much candy

would they get each?”

Let the students work out their answers with Unifix cubes.

Record the result as: 6 ÷ 5 = 1 whole and 2 tenths.

Have the students perform the calculation 6 ÷ 5 = on a calculator and compare the

result with the candy bar sharing.

Provide other examples like this to consolidate the link between the calculator

displays and the number of whole bars and tenths.

Examples might be:

“Four candy bars and five people” (4 ÷ 5 = 0.8)

“Five candy bars and two people” (5 ÷ 2 = 2.5)

“Six candy bars and four people” (6 ÷ 4 = 1.5)

Change the problems to addition and subtraction. This will require the students to link

the numbers on both sides of the decimal point.

Problem: “Henry has 3.4 candy bars, and Tania has 1.8 candy bars.

Peter thinks that they will have 4.12 bars altogether. Use the Unifix cubes and

calculator to find out if he is right.”

The students should find that the 0.4 and 0.8 combine to form another whole bar and

two-tenths, and so the answer is 5.2. The calculator will confirm this.

The students can record their ideas in this format:

With each problem, the students should model the problem with the Unifix cubes

first, predict the correct answer, and confirm it on a calculator.

Problem: “Jody has 4.3 candy bars. She thinks that if she eats 2.7 candy bars, she will

have 2.4 left. Is she right?”

Problem: “Rangi-Marie thinks that 7 x 0.4 candy bars will be 0.28. Is she right?” This

means seven lots of point four candy bars.

Problem: “Timoti thinks that 5.2 ÷ 4 = 1.3 (5.2 bars shared among four people). Is he

right?”

### Using Imaging

*Shielding*: Set up similar problems by masking the Unifix cube model under sheets of

paper or ice-cream containers. For example: “What is 5 x 0.6?” Use five ice-cream

containers, each holding a stack of six cubes (six tenths). Label the top of each

container with 0.6.

Discuss why the answer is 3.0 and not 0.30 (Thirty tenths is the same as three ones).

Similar problems might be:

3.7 + 2.6 = (Two containers holding 3.7 and 2.6 stacks respectively)

5.1 – 2.9 = (One container holding 5.1 and another upturned into which 2.9 will be

put)

7.2 ÷ 6 = (7.2 visible to be shared among six containers)

Extend the students’ strategies into hundredths by posing problems like:

“There are five candy bars and four people. How much candy will they get each?”

The students will realise that each person will get one whole bar, but sharing the

remaining bar among four cannot be done just using tenths. Two tenths can be given

to each person, but two tenths remain.

The students may suggest that each person can then have half of a tenth. Point out that

our number system works on breaking and making into tens.

Ask, “If I break one-tenth (holding a single cube) into 10 equal pieces what will they

be called?” (hundredths) “How many hundredths will each person get in our

problem?” (five)

Record the answer as: 5 ÷ 4 = 1 whole + 2 tenths + 5 hundredths

Perform 5 ÷ 4 = 1.25 on a calculator and ask the students to explain the display. Pose

similar problems involving hundredths like:

“Three bars shared among four people” 3 ÷ 4 = (0.75)

“Seven bars shared among five people” 7 ÷ 5 = (1.4)

13 ÷ 4 = (3.25) 9.6 ÷ 8 = (1.2)

Where necessary, Unifix cubes can be used to model these problems.

### Number Properties

Pose operational problems that involve working with ones, tenths, and hundredths

directly and require exchanging across the decimal point.

2.4 + 4.57 = (6.97) 6.53 – 2.7 = (3.83) 3.4 – 1.29 = (2.11)

7 ÷ 4 = (1.75) 1.25 x 6 = (7.5) 5.3 ÷ 5 = (1.06)