C1.4 Create and describe patterns to illustrate relationships among whole numbers and decimal numbers.
Activity 1: Number Patterns
In this activity students are developing their algebraic reasoning by exploring number relationships using a calculator.
Give an instruction such as, "Press [1], [x], [10], and [=] or [1 000 000], [÷], [10], and [=]. You will see the number 10 or 100 000 displayed."
Have students uncover the relationship between the numbers displayed by asking questions such as:
- Can you predict the next number before you press the [=] key?
- What will happen if you press the [=] key three more times?
One by one, write down all the number patterns that students have explored on the calculator.
- 10, 100, 1000, 10 000, 100 000, 1 000 000
- 1 000 000, 100 000, 10 000, 1000, 100, 10, 1, 0.1, 0.01, 0.001
Ask students to observe what happens to the quantity between each term in each of the patterns, (a) and (b). The next term is always 10 times greater in the pattern shown in (a) and 10 times less in the pattern shown in (b).
Source: adapted and translated from Guide d’enseignement efficace des mathématiques de la maternelle à la 3e année, Modélisation et algèbre, Fascicule 1, Régularités et relations, p. 66.
Activity 2: Decimal Numbers Relationships
Present the students with the following two sets of numbers:
Set 1:
\(3.425 = 3 \ \mathrm{ones} + 4 \ \mathrm{tenths} + 2 \ \mathrm{hundredths} + 5 \ \mathrm{thousandths}\)
\(3.425 = 3 \ \mathrm{ones} + 4 \ \mathrm{tenths} + 1 \ \mathrm{hundredths} + 15 \ \mathrm{thousandths}\)
\(3.425 = 3 \ \mathrm{ones} + 4 \ \mathrm{tenths} + 25 \ \mathrm{ thousandths}\)
Set 2:
\(\displaylines{\begin{align} 5.430 + 0.011 = 5.441 \ \ & \ \ 5.441 \ - \ 0.011 = 5.430 \\ 5.431 + 0.010 = 5.441 \ \ & \ \ 5.441 \ - \ 0.010 = 5.431 \\ 5.432 + 0.009 = 5.441 \ \ & \ \ 5.441 – 0.009 = 5.432 \end{align}}\)
a) Look at these two number sets and describe the relationship between the numbers in each set.
Sample Responses
Set 1:
Set 1 shows the relationships between the place values of the base-ten number system and its multiplicative rule of 10.
Set 2:
Set 2 shows the connection of adding and subtracting thousandths to a number. To get the same results, I have to decrease the 2nd term of the addition by 1 thousandth when I add 1 thousandth to the 1st term. In subtractions, when I subtract 1 thousandth less, the difference gets 1 thousandth more.
b) It is your turn to create a set of numbers that involves an operation. Describe your pattern, and explain the relationship between the numbers in your pattern.
Here is my set:
\(\displaylines{\begin{align} 1 \times 0.002 &= 0.002 \\ 2 \times 0.002 &= 0.004 \\ 3 \times 0.002 &= 0.006 \\ 4 \times 0.002 &= 0.008 \\ 5 \times 0.002 &= 0.0010 \\ 6 \times 0.002 &= 0.0012 \\ \mathrm{…etc.} \end{align}}\)
I multiply 0.002 by 1, then by 2, by 3, etc. I show that we can apply our multiplication (and division) number facts by making sure to put the thousandths correctly with the respective positions.
Source: translated from En avant, les maths, 6e année, CM, Algèbre, Habiletés liées aux relations dans les suites, p. 10-11.