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## Haasteiden selitykset:

### Desimaalien merkitys

### Suhteellisuus

Oppilaalla vaikeuksia desimaalien vaikutuksen ymmärtämisessä.

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## Equivalence

The student does not understand that a rational number can be represented in many ways (by an infinite number of terms; e.g. 0.5=0.50=0.500=1/2=2/4=3/6=...). The student does not perceive the connection between fraction and decimal numbers.

**Tip 1:**Study the equivalence by using area models and number lines. Ask the students to draw for example a half in several ways.

**Tip 2:**You can explore equivalence of fractions by forming fraction numbers out of students. Tape a division line on the floor. Ask a student to go above the line as a nominator and two students to go below the line as a denominator (fraction 1/2). Write 1/2 on the whiteboard. Continue by adding students to the nominator and to the denominator in the same ratio: 1 student to the nominator, 2 to the denominator. Always write the formed fraction on the whiteboard as a symbol. Discuss with students whether the ratio of the students above and below the division line changed. Also convert the fraction numbers on the board to decimal numbers.

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## Understanding a fraction

The student does not understand that the fraction magnitude is represented as a ratio between the two terms (nominator divided by denominator).

- Students may view fractions as two separate integers. When comparing fractions a student might think that the larger fraction can be determined by comparing the nominator and the denominator separately. For example, the student might falsely think that 2/9 is larger than 1/3, because 2>1 and 9>3.
- The student might utilize a comparison strategy where he or she systematically assumes that the largest fraction number is the one that is missing the least pieces. For example the student might falsely think that 1/2 is larger than 3/5 because it is only missing one piece (2-1=1) as 3/5 is missing two pieces (5-3=2).

**Tip:**Visualize fraction magnitudes by drawing three candybar comparison tasks: a) 1/3 vs 2/3, b) 2/9 vs. 1/3, c) 1/2 vs. 3/5. Emphasize that the size of the candybar (magnitude) cannot be estimated based on the remaining pieces. Instead the remaining pieces must be compared to all the pieces of a whole candybar. Also show that missing pieces strategy does not always work and thus it shoudn't be used.

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## The significance of decimals

The value of a decimal number cannot be derived from the number of digits.

decimal numbers do not followthe pattern that (1.65 is smaller than 1.7), as is the case with natural numbers.The student may falsely assume that more digits indicate a larger magnitude (i.e. 0.6 is smaller than 0.12). Failures with the comparison of decimal numbers may also be due to students’ component based comparisons of the terms before and after the decimal (as in 12 > 6, therefore 0.12 > 0.6).

**Tip 1:**Discuss this in the context of money. Consider one euro as a whole which consists of hundred cents. Consider whether a lollipop costing 12 cents is more expensive than a lollipop costing 0.6 euros i.e. 60 cents.

**Tip 2:**Study this by drawing two grids with both containing hundred squares. Emphasize that an entire grid equals a whole. Color tenths (a whole column) with a different color than hundreths (an incomplete column). Compare the values based on the colored grids. Additionally the same should be demonstrated using a number line.

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## Density of rational numbers

The student has difficulties understanding that there are infinite number of rational numbers in between any two non-equal fraction or decimal numbers. Although with whole numbers it is always possible to distinguish the successor number of any number, it is not possible to distinguish the successor number of a fraction or a decimal number. The student might falsely think that for example 0.3 comes straigth after 0.2 and there are no other numbers in between because based on whole number properties the student assumes that after number 2 comes number 3. In a similar fashion the student might think that for example 3/4 comes straigth after 2/4 with no numbers in between.

**Tip 1:**Draw a number line to the whiteboard with end values 0.5 and 0.6. Ask the student what decimal numbers are there between these numbers. Add the numbers to the number line. Continue until you run out of space. Then you can explain that there is an infinite amount of numbers and there is no way you will be able to place them all to the whiteboard. Draw another number line below with end values 1/2 and 2/2. Ask the students what fraction numbers exists between these numbers and place them to the line. In the end you can demonstrate how the decimal numbers are connected to the lower number line ranging from 1/2 to 2/2.

**Tip 2:**Study the density concept using the zooming metaphore. Draw a number line to the board. Select a part of the number line (zooming in). Draw another number line for this part then add some numbers to it. Keep zooming by selecting a new part and progress deeper and deeper. Do this with fraction and decimal numbers as well as combinations.

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## The significance of zeroes

The student does not understand when a zero has a significance to the value of the decimal number and when not.

- The student might believe that a zero following the decimal delimiter is meaningles. Therefore he or she might assume that 0.05 is equal to 0.5.
- The student might assume that 0.50 is larger than 0.5. He or she does not undestand that the zeroes in the end of the decimal number do not change the value, but may assume that 0.50 is larger while it contains more digits.

**Tip 1:**Think this with money example. Which one costs more: a lollipop costing 0.05 euros or a lollipop costing 0.5 euros?

**Tip 2:**Study this by drawing two grids with both containing hundred squares. Color tenths (a whole column) with a different color than hundreths (an incomplete column). Increse the hundreths of one of the grids gradually so that the student can see visually how the value increases towards the next tenth. Additionaly you can demonstrate the difference of sizes on a number line.

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## Student details