Promoting Number Sense: Beyond Computation

Carlo Magno, PhDDe La Salle University, Manila

Target Standard in the K to 12 curriculum

Grade 5 Performance Standards read and write large whole numbers and

round off whole numbers to the nearest thousands and millions.

find the greatest common factor and the least common multiple of given numbers.

apply divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 11 and 12 on different contexts.

simplify a series of operations on whole numbers and solve problems involving these.

Grade 5 Performance Standards perform the four fundamental

operations on fractions and mixed numbers and solve related problems

investigate the relationship between fractions and decimal numbers.

explore, know and understand the concept and value of a decimal number.

add and subtract decimal numbers with values through thousandths and solve problems involving these.

Grade 5 Performance Standards multiply decimal numbers of values up to

the hundredths and solve problems involving these numbers.

divide decimal numbers of values up to the hundredths and solve problems involving these numbers.

manipulate ratios and solve problems involving ratios and proportions

know and understand the concept of percent and to solve problems involving percents.

Grade 6 Performance Standards explore the concept of integers and compare

integers with whole numbers, fractions and decimal numbers.

use the notation and computation of positive whole number powers to find values of numbers.

perform the four basic operations on integers perform a series of operations following the

GEMDAS rule and solve problems involving a series of operations on whole numbers.

Grade 6 Performance Standards add and subtract fractions and decimal

numbers and solve problems involving these numbers.

multiply fractions and decimal numbers and solve problems involving multiplication of fractions and decimal numbers.

divide fractions and decimal numbers and solve problems involving division of fractions and decimal numbers.

Grade 6 Performance Standards

use ratio and proportion in a variety of ways and solve problems in different contexts.

use percent in a variety of ways and solve problems in different contexts.

Watch a video on developing number sense

Observe the answer of the student

96- 24 72

14- 9 15

21- 16 15

What do you think is the problem in the child’s concept in subtraction?

What is number sense?

Number sense is the ability to understand and use numbers and operations flexibly.

Number sense results in a view of numbers as meaningful entities and an expectation that mathematical manipulations and outcomes should make sense.

What is number sense?

Number sense involves: understanding numbers knowing how to write and represent numbers in different

ways recognizing the quantity represented by numerals and

other number forms discovering how a number relates to another number or

group of numbers Number sense develops gradually and varies as a

result of exploring numbers visualizing them in a variety of contexts relating to them in different ways

What is number sense?

In the primary and intermediate grades, number sense includes skills such as: counting representing numbers with manipulatives and models understanding place value in the context of our base 10

number system writing and recognizing numbers in different forms such

as expanded, word, and standard expressing a number different ways—5 is "4 + 1" as

well as "7 - 2," and 100 is 10 tens as well as 1 hundred ability to compare and order numbers—whole numbers,

fractions, decimals, and integers the ability to identify a number by an attribute—such as

odd or even, prime or composite-or as a multiple or factor of another number

Expression of quantities in the real world

Examples Whole numbers describe the number of students in a

class or the number of days until a special event. Decimal quantities relate to money or metric

measures Fractional amounts describing ingredient measures or

time increments Negative quantities conveying temperatures below

zero or depths below sea level. Percent amounts describing test scores or sale prices. Able to estimate and make a meaningful

interpretation of its result.

Prerequisite skill

Number sense develops as students: understand the size of numbers, develop multiple ways of thinking about

and representing numbers, use numbers as referents, develop accurate perceptions about the

effects of operations on numbers (Sowder 1992).

What do we need to assess in number sense?

Using numbers flexibly when mentally computing, estimating

Judging magnitude of numbers and reasonableness of results

Moving between number representations Relating numbers and symbols and

operations Making sense of numerical situations Looking for links between new information

and previously acquired knowledge

Teaching Approach

There should be a balance in teaching mathematical algorithms with mental computations, problem solving and the use of calculators.

Problem-centered projects show how children who engage in meaningful arithmetic problem solving also develop autonomy and adaptability in the learning situation.

What types of strategies do children develop?

Partitioning numbers using tens and ones. "First I added the 20 and 10 and got 30. Then I added the 9 and 4 and got 13. Then I added the 10 from 13 to 30 and added 3 more and got 43.

Counting on or back from a number. "First I counted on from 29 by tens and went 29, 39. Then I counted on 4 more — 40, 41, 42, 43."

What types of strategies do children develop?

Using "nice numbers." Nice numbers are multiples of 10 or other numbers that are easy to work with. "I know that 30 plus 15 is 45, but 29 plus 14 is 2 less than that, so it’s 43."

Translating to a new problem. "I took one away from the 14 and gave it to the 29 to make 30. Then I had 30 plus 13, which is 30 plus 10 plus 3, which is 43.

Understanding Processes

Mental computation written computation using calculators

Mental Computation

Mentally estimates and calculates addition and subtraction to 100 using strategies based on ones and knowledge of number facts.

Use alternative strategies to decompose (break up) two-digit numbers to add and subtract numbers, ex. “to subtract 43 I took off the 40 and then 3 more’, or ’26 and 43 is 43, 53, 63 and 6 more is 69’.

Mental Computation

Adds a list of one-digit numbers or amounts in an efficient way, e. g. 76c + 49c +14c … 76c + 14c makes 70, 80, 90 plus 50c is Php 1.40 less 1c makes Php 1.39.

Multiplies and divides by one-digit numbers involving multiples of 10.

Written Computation

Uses a variety of strategies to add and subtract whole numbers and money only when mental strategies are inadequate.

Uses regrouping in number exploration and in computation such as in showing decomposition (346 is 34 tens and 6 ones or 3 hundreds and 46 ones ) or when needing to subtract (46 ones becomes 3 tens and 16 ones).

Written Computation

Uses estimation and approximation to round off and justify whether sums and differences are sensible.

Uses multiplication and division as inverse operations and recognize the commutativity or multiplication (36 9 = 4, 4 x 9 =36.

Written Computation

Shows understanding of multiplication and division based on equal groups and uses written methods for calculating products and quotients such as expanded multiplication, ex. 345 x 3 is 300 x 3 40 x 3 5 x 3, and division, such as 345 3 as 3 x ? makes 300 3 X ? makes 45.

Devise shorter methods for written calculation based on understanding of strategies based on groupings.

Calculator

Uses a calculator appropriately by entering digits correctly and in the right order for all operations.

Makes estimates and checks reasonableness of estimates.

Checks whether a prediction about numerical operations or patterns holds true when computational demands are beyond the child’s levels of ability, ex. 15 + 15 = 130, so 115 + 115 = 230, so 1115 + 1115 = 2230.

Calculator

Tests understanding of place value such as increasing digits by ten or pretending that a numerical key is inoperable.

Finds relationships between operations such as 45 X 6 is the same as 45 + 45 + 45 +45 + 45 + 45.

Finds related patterns in the number system by exploring the effect of multiplying and dividing by 10 including decimals.

Explores rounding-off options.

Numerical Relationships

Traditional

Product of 12 and 6 is _____12 X 6 =6 X 12 =120 X 60 =12 X 0.6 =Complete the pattern: 6, __, 18, __, 36List factors 0f 72 ______

Numerical Relationship

1. Make up a number story that matches 12 x 6 =. Explain how you got the answer. Think about how your answers would change if you changed the problem to 120 x 6. Would the answer be smaller or larger if it was changed to 120 x 0.6? Explain your thinking by recording your solutions.

Numerical Relationship

2. Is 12 x 6 the same as 6 x 12? Explain your thinking by recording your solution.

3. Why is 12 6 different from 12 x 6? Explain or thinking. Make up a problem to show how the two number facts are different.

Teaching Number Sense

Teaching with manipulatives and models

Graphic representations Daily Routines Games

Teaching with manipulatives and models

place-value blocks fraction strips decimal squares number lines place-value and hundreds charts

Graphic representations

Use of frames How many counters are there? How many spaces without counters

are there?

Watch video on ten frames

Graphics Representation Hundreds chart (121)

•Help children recognize number relationship•Work with large numbers (estimation)•visual support for problem solving strategies

Daily Routines

Many teachers use the calendar as a source of mathematics activities. counting, patterns, number sequence, odd and even

numbers, and multiples of a number create word problems related to the calendar

A hundreds chart can help them count the number of days in school, and the current day’s number can be the "number of the day.

On the 37th day of school, children may describe that number as 30 plus 7, 40 minus 3, an odd number, 15 plus 15 plus 7, my mother’s age, or 1 more than 3 dozen.

Games

The calculator can be a fun tool for quick mental computation practice.

Have the children predict the sum or difference and the calculator is used to confirm.

Games

Press 0 + 2 = on the calculator. Then press any other number, such as 7,

and hold your finger over the = key. Have the children predict the sum then

press the = key to confirm it. As long as they continue to press only a

number and =, the "machine" will continue to add 2.

The use of the calculator and the immediate feedback reinforce computation and encourage children to keep playing

Games

Guess My Number Choose a secret number and tell children a range that

your number falls within. You can start small, with 1 to 10, or use a larger range

(such as 1 to 100, 25 to 75, or 150 to 250). Have children guess your number and tell them

whether their guess is larger or smaller than your number.

Children will quickly develop strategies that help them zero in on your secret number.

To extend this game, choose a secret number from a wide range such as 1 to 500 and give one clue, such as that it is even, it ends in 4, or the sum of the digits is 9; then ask students to start guessing.

Games Stand Up and Be Counted Ask children to describe the number 25 in as many

ways as they can ("number of the day,“) and record their ideas as an example.

Each child draw from a bag of squares numbered 1 through 100 and write down as many ways as they can to make the number they drew.

Ask a volunteer to stand up and read one statement at a time about his or her number.

If that statement is true for other children’s numbers, they stand up.

If it is not true, they remain seated. Through discussion, the children can begin to focus

on the characteristics of the numbers and their relationships

Stretch Students' Thinking

Help students identify whole-number relationships that are different from decimals, fractions, and integers.

Students may unsuccessfully try to apply these relationships to decimals, fractions, or integers.

34.5 > 3.456, 1.11 < 1.111, 68.2 = 68.20

Stretch Students' Thinking

The whole number 6 is greater than the whole number 5, but when unit fractions have these numerals as denominators, the relationship is reversed, and 1/5 > 1/6. Similarly, -5 > -6

Applications

Reading/English Read and discuss any of the many available

counting books that illustrate numbers up to 10, 20, and so on.

Identify ways that numbers are represented in print. ▪ "When are numbers shown in standard form? ▪ Word form? ▪ Short-word form? ▪ When are actual numbers used? ▪ When are rounded numbers used?"

Applications

Writing Ask students to write about number

representations by defining and giving examples of different forms of numbers-standard, expanded, word, and short-word.

Applications

Social Studies For students in primary grades, find and

discuss different ways numbers are used in the environment (e.g., addresses, time, temperatures, grades, speed limits, phone numbers, on recyclable plastics).

Have students in intermediate grades find population figures for town or city, state, and country. Then ask them to compare and order the populations they found with those other students found.

Applications

Science represent numbers with concrete objects. Choose linear measurements relating to

science, such as the sizes of dinosaurs, and represent lengths using pieces of string or yarn.

Label the strings, and then compare and order them.

Make comparisons between string lengths, string length and classroom dimensions, string length and students' height, and so on.

Exercise

Give examples of games to enhance number sense

Applications

Discuss and practice writing numbers using scientific notation. Have them find examples of measures written with scientific notation and identify the situations in which they are used and why.