C1.4 Create and describe patterns to illustrate relationships among whole numbers and decimal tenths and hundredths.
Activity 1: Counting in choir
From a young age, students count in increments of 1, 2, 5, 10, 25, etc. Counting by leaps and writing the resulting number sequence helps students see the relationships between numbers. It is important to continue to have students count in the junior grades with different types of numbers, such as large natural numbers (e.g., counting by leaps of 5,000, 25,000), fractions (e.g., counting by leaps of "one-third" by saying 1 one-third, 2 one-third, 3 one-third, 1 whole, etc.), and so on, up to at least 5) and decimal numbers (tenths or hundredths). The way numbers are written when students recite them is intentional and should be planned to show relationships.
Prerequisite
Before having students count in "two-tenths" increments, lead a mini-lesson counting in "one-tenth" increments.
Materials
- whiteboard or chart paper
- felt-tip pens
- number line 0 to 5 divided into tenths or 10 frames (explain that for the activity each frame is considered 1 whole)
Approach
- Gather the students and tell them that they are going to count together in "2 tenths" increments from 0 to 5. First, give the students a minute to think about the numbers that will be recited. Then, begin counting in a chorus at a pace that allows you to write the numbers. You can also show the visual on the number line or on the 10 frames.
- As students recite the numbers, write them down like this: 0.2 0.4 0.6 0.8 (two-tenths, four-tenths, six-tenths, eight-tenths).
- Pause here and ask students to think of the next number. Ask them how they will pronounce it and how it will be written.
- Conduct a Think-Pair-Share.
- Note that a number of students will think that the next number will be written 0.10.
- As you pronounce 10 tenths \(\frac{10}{10}\), write 1. Ask students why it is spelled that way. So, 10 tenths is ONE. This is best illustrated on the number line or with a filled-in 10 frame.
Sequence of numbers starting at zero point 2 and continuing to 5 in leaps of plus zero point 2.
0,2 0,4 0,6 0,8 1
Continue to 5, writing the numbers in this way to highlight the relationships between them.
0,2 0,4 0,6 0,8 1
1,2 1,4 1,6 1,8 2
2,2 2,4 2,6 2,8 3
3,2 3,4 3,6 3,8 4
4,2 4,4 4,6 4,8 5
Ask students to observe the written numbers after counting in two-tenths increments to 5.
Ask them what they notice about numbers by conducting a Think-Pair-Share.
Point out the relationships between the numbers they observed. Here are some examples:
Sequence of numbers starting at zero point 2 and continuing to 5 in leaps of plus zero point 2. The numbers are placed in 5 columns, all the comma of 2's are placed together and are circled. All the decimal 6 are placed together and are circled. All whole numbers are placed together and circled.A numerical band from zero decimal zero to zero decimal 8. Arrows represent leaps of plus zero decimal 2.
Suggestion
Use this opportunity to make the connection with fractions. For each decimal number recited, you can also write a fraction. For example, 0.2 can be written as 2/10. Lead students to understand that decimal numbers in tenths can also be written as fractions with a denominator of 10. Both represent the same quantity.
Similarly, if you are counting in hundredths, take the opportunity to make the connection between decimal numbers and fractions with 100 as the denominator. For example, it is possible to write 25 hundredths as 0.25 or 25/100.
Activity 2: Relationships in decimal numbers
Present the following situation to students:
a) Describe Miray's patterns showing the relationship between ones, tenths, and hundredths.
| 4.63 |
|---|
| 4 units + 6 tenths + 3 hundredths |
| 4 units + 5 tenths + 13 hundredths |
| 4 units + 4 tenths + 23 hundredths |
| 4 units + 3 tenths + 33 hundredths |
| 4 units + 2 tenths + 43 hundredths |
| 4 units + 1 tenth + 53 hundredths |
| 4 units + 0 tenths + 63 hundredths |
| 4.63 |
|---|
| 4 units + 6 tenths + 3 hundredths |
| 3 units + 16 tenths + 3 hundredths |
| 2 units + 26 tenths + 3 hundredths |
| 1 unit + 36 tenths + 3 hundredths |
| 0 units + 46 tenths + 3 hundredths |
Strategy 1
Describing patterns using words
- In this pattern, I notice that as the tenths decrease by 1, the hundredths increase by 10. In each tenth, there are 10 hundredths.
- In this pattern, I notice that as the ones decrease by 1, the tenths increase by 10. In each one, there are 10 tenths.
Strategy 2
Describing a pattern using base 10 material
- I represent the first 4 equalities in the 1st set of related operations using base 10 material. In each tenth, there are 10 hundredths. I notice that as the tenths decrease by 1, the hundredths increase by 10. A representation that show blocks of hundredths, sticks of tens, and cubes of a unit. It is mentioned that in the represented step: A hundred block is equal to one. 6 tenths becomes 5 tenths. 5 tenths becomes 4 tenths. 4 tenths becomes 3 tenths. 3 hundredths becomes 13 hundredths. 13 hundredths becomes 23 hundredths. 23 hundredths become 33 hundredths.
- I represent equalities in the 2nd set of related operations using base 10 material. In each unit, there are 10 tenths. I notice that as the units decrease by 1, the tenths increase by 10. A representation that shows us blocks of hundreds, sticks of tens and cubes of, a unit. It is mentioned that in the represented approach: -One block of hundreths is equal to one. - 4 units become 3 units. - 3 units become 2 units. -2 units become one unit. -6 tenths becomes 16 tenths. -16 tenths becomes 26 tenths. -26 tenths becomes 36 tenths. -36 tenths becomes 46 tenths.
b) Create a set of related addition operations and a set of related subtraction operations using the number 4.63.
c) Miray wonders if there are any patterns in the multiplication and division tables. Create a numeric pattern to show the relationship between multiplication and division facts for a number of your choice.
Source : En avant, les maths, 4e année, Algèbre, p. 9-13.