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Decimal Concepts Although students begin to develop place value understandings in primary one, they are formally introduced to decimal numbers usually from primary four. 10 Ed uc 2 5 at io n By the time students reach primary four, they would have been introduced to both whole number place value and fractions. Hence, at this point we will assume that students have some knowledge of whole number place value as well as of fractions 1 1 1 to include at least , , and . ✎ Let us consider these place value and fractional prerequisites in more detail. al lC av en di As with most topics, effective mathematics place value instruction begins with using some relevant concrete material. As highlighted in Figure 1, let us assume that students became familiar with base ten blocks. At some point the teacher compared the actual blocks to a picture of the base-ten blocks perhaps in a textbook or a PowerPoint slide. Likely, the teacher would have stated the value of those blocks and eventually written the value on the whiteboard. M ar sh Concrete Number symbol 23 Word Twenty three Figure 1 Concrete-pictorial-abstract inter-relationships for place value concepts for whole number Place value Teaching to Mastery Mathematics: Teaching of Decimals sh Place Value 5 (M)TTMM_Decimals_01.indd 5 2/4/13 11:06 AM Concept of Tenths at io n With prerequisites in place, students will likely be presented with experiences similar to those shown in Figure 2 and followed by instruction to highlight key points beginning with decimal ideas as suggested by Figure 3. Ed uc I folded the strip of paper into 10 equal parts. Divide 1 whole into 10 equals parts. 1 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 We can write 1 tenth as 1 sh 0.1 is 1 tenth. M ar We write 5 tenths as 0.5. We read 0.5 as zero point five. Notes 0.1 and 0.5 are decimals. These are decimal points. Figure 3 Writing tenths in decimal form 5 or 0.5. 10 1 or 0.1. 10 Teaching to Mastery Mathematics: Teaching of Decimals al lC av Figure 2 Modeling tenths en di sh I coloured one tenth of the rectangle. 7 (M)TTMM_Decimals_01.indd 7 2/4/13 11:06 AM ✎ Pedagogy-based Task How might you use the number line to reinforce mixed-number decimal concepts? • Give number; find number on the at io n Three illustrations are given. For each one, what additional questions would you consider asking? • Use the number line to show the number 2.6. number line • Show point on number line; name the number 1 2 3 Ed uc 0 • Find the value of Points A and B on the following number line. A 1 2 3 sh 0 B sh al lC av Questions: What is 0.1 more than 0.6? What is 0.1 less than 0.6? What is 0.1 more than 0.9? What is 0.1 less than 0.9? 0 1 2 Develop one or two other illustrations that also make use of the number line. M ar Chapter 1: Developing Decimal Number Concepts number line en di • Compare numbers using the • Use the following number line to help you answer the questions. Solution on p.34 10 (M)TTMM_Decimals_01.indd 10 2/4/13 11:06 AM Number Line The number line, although perhaps not as “concrete” as some other illustrations, has a distinct advantage of allowing representations of tenths, hundredths and thousandths. At some point, a unit has to be established. Then within that unit if necessary a pair of end points must be chosen so that subsequent discussion of points between those points can be made. The diagram and examples below help show how units and sub-units can be modeled. at io n ✎ number line 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 1 Ed uc 0.1 0.30 sh 0 EXAMPLE 1 al lC av In the first set of examples below, although some concentration is required, students are basically asked to read the numbers off the number line. M ar sh What decimal does each letter represent? A B C a) 0.01 b) 0.02 J 9.97 0.03 K 9.98 0.04 L 9.99 M 10.00 10.01 Solution on p.34 Teaching to Mastery Mathematics: Teaching of Decimals en di As was suggested with the activity with number discs, number lines can also be used to promote various levels of conceptual understanding. 21 (M)TTMM_Decimals_01.indd 21 2/4/13 11:07 AM