DECIMAL NUMBER SENSE

MaTHink 2015

Riverside County Office of Education

Susan AddingtonCSUSB

Why are decimals hard?

Place value and

Fractions

Is place value hard?

Easy questions: What is the place value of 7 in 2709? What’s a quick way to multiply by 10?

Harder questions: How many millions are in a billion? Suppose place values went by 5’s instead of

by 10’s. What is the place value of 3 in 2341? Sam started work at 2:48 and ended at 9:14.

How long did she work? Do the subtraction:

Do the subtraction

Hint: what is the place value of the 1 in 9:14?

The key idea of place value

The value of each place is 10 times the value to its right, and 1/10 the value to its left.

Requires consistent thinking about “10 times as big”, “1/10 as big”.

Iterated multiplication and division, NOT addition.

Research in grades 3-5

Researcher: How many toy wheels are in this bag? Two clues:

There are enough for six toy cars. There are 2 left over.

Students: 26 wheels. Researcher: What does each part of 26 (the 6 and then the

2) have to do with “how many you have”?

What do you think students said?

Ross, S. R. (2002). Place Value: Problem Solving and Written Assessment. Teaching Children Mathematics, 8(7), 419–23

Student work

Moral of the story

Don’t assume your students completely understand place value.

Like many mathematical topics, full understanding develops over many years.

4th grade standards related to decimals

Understand decimal notation for fractions, and compare decimal fractions (including using a number line)

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; … l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit.

[3rd grade:] Square units such as square cm

Everyday objects

What could you easily divide into tenths to help students understand decimals like 1.4?

A peanut butter sandwich Do we have enough so that everybody can have .3 of a

sandwich? An apple: cut in half horizontally; use seeds to cut

each half in fifths. How much would each person get if we shared these 12

apples equally with everyone, and we cut the apples in tenths? Cut the apples, give out, check with a calculator.

Money How much is 1/10 of a dollar? What fraction of a dollar is a penny?

Fraction strips

Tenth strips and hundredth strips

(Available soon, I hope, from the Measuring the World store at http://www.quadrivium.info)

Cuisenaire rods

Lengths are 1 cm to 10 cm.

Use the 10 cm rod (orange) as the whole.

What numbers are shown by the other lengths?

Standard units in the metric system Make sure everyone has experience measuring

length in meters and centimeters Meter sticks (at least one per group) Rulers Metal tape measure: buy in bulk at Dollar Tree.

Chinese-made ones have both inches and cm. Prefixes:

Deci- means 1/10 of Centi- means 1/100 of Milli- means 1/1000 of Kilo- means 1000

Find these numbers on a meter stick or tape measure

a. 0.4 metersb. 0.38 metersc. 0.97 metersd. 0.06 meterse. 0.003 metersf. 1.468 meters

How would you say these same lengths in centimeters?

In millimeters?

Measure in decimal meters early and often!

How long is the table, in meters? How long is the room? How wide is the white board? How tall are you? …

Cooking

Don’t need to use a stove; make non-cook recipes with a few ingredients Fruit salad Punch with several kinds of juice and fizzy water Salads with grains and chopped vegetables

Most liquid measuring cups have both US and metric units. (Go back to the dollar store!)

Measure in decimal liters (and/or milliliters) If you have cooking scales, measure in

kilograms (and/or grams)

Base ten blocks

Smallest piece: a centimeter cube (“small cube”)

Dimensions: 1 cm by 1 cm by 1 cm. Volume: 1 cubic centimeter

“Long” Dimensions: 10 cm by 1 cm by 1

cm. Volume: 10 cubic centimeters,

10 small cubes

Base ten blocks

“Flat” Dimensions: 10 cm by 10 cm by 1

cm. Volume: 100 cubic centimeters, 100

small cubes, 10 longs

Big cube Dimensions: 10 cm by 10 cm by 10

cm. Volume: 1000 cubic centimeters,

1000 small cubes, 100 longs, 10 flats

Base ten blocks too expensive to make

A big long. How many big cubes? How many flats? How many longs? How many small cubes? Dimensions? Volume in cubic

centimeters?

Base ten blocks too expensive to make

A big flat. How many big longs? How many of [change

name of piece]? Dimensions? Volume in cubic

centimeters?

Base ten blocks too expensive to make

A huge cube. How many big

flats? How many of

[change name of piece]?

Dimensions? Volume in cubic

centimeters? In cubic meters?

Base ten blocks too expensive to make

Susie SketchUp

A huge long.How many huge cubes?How many of [change name of piece]?Dimensions?Volume in cubic centimeters?

Base ten blocks too expensive to make

A huge flat. Same questions as

before.

Base ten blocks too expensive to make

A giant cube.

Same questions as before.

Do not call the small cube “the unit” This suggests that its size (volume, or

area covered) is 1. Suppose the flat is 1. What is the volume of the long? 1/10, or .1 What is the volume of the small cube? 1/100, or .1 What is the volume of the big cube? 10

On this page, the flat = 1

How would you show these numbers with blocks?

a. 0.34b. 0.123c. 0.007d. 3.568e. 0.020

Change the unit again

What would you choose as 1 in order to show these numbers? You can choose a different piece for 1 for each number.

a. 45.29b. 0.00005c. 0.0864d. 245,700,000

5th grade standards related to decimals

Understand the place value system (including decimals to thousandths)

Perform operations … with decimals to hundredths.

Convert like measurement units within a given measurement system.

Cubic units (such as cubic cm)

6th grade standards related to decimals

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Adding and subtracting decimals with base ten blocks

Prerequisite: add multi-digit whole numbers with base 10 blocks.

Whole numbers: small cube = 1, long = 10, flat = 100

Decimals: choose a unit as appropriate.

Try it.

Do these additions with base ten blocks. For each, specify the place value of each

block. Explain what happens when you

regroup/trade.a. 2.46 + 1.57b. 0.278 + 0.394c. 32.1 – 4.7d. 0.42 – 0.19e. 0.00000068 – 0.00000032

Make sure to connect written work with blocks.

Money?

I have $3.15. I owe Frida $1.78. How much will I have left after I pay her back?

Act it out with two people.

Multiplication

Prerequisites: The area model of multiplication: to show

AxB=C, use a rectangle whose side lengths are A and B. Then the area is C. (CCSS 3rd grade.)

3x4 is 3 rows of 4 squares or 4 columns of 3 squares.

Area and multiplication

Use inch rulers and square inch tiles.

6 inches x 8 inches = 48 square inches

Multiplication

Another prerequisite: Multiplying 2-digit whole numbers with base

ten blocks. (Appropriate for 4th grade).

Multiplying to tenths

Materials: Rulers in inches and tenths (from Measuring

the World store soon?) Graph paper in inches and tenths (from

http://incompetech.com/graphpaper/ Square inch tiles

Explore the materials

Look carefully at the ruler, and at some ordinary purchased ruler. What size is the smallest space between marks? How do you know?

Explore the materials

Look carefully at the graph paper.

What are the side length and area of a large square?

What are the side length and area of a tiny square?

What fraction of a large square is a row of 10 tiny squares?

2.3 inches × 3.6 inches

On the graph paper, Measure the base

of a rectangle 1.3 inches

Measure the height of the rectangle 4.2 inches.

Carefully draw it. What is its area,

in square inches?

Zoom in

2.3 inches × 3.6 inches = ? How big are the tiny squares on the

graph paper? Answer:1/100 of a square inch=0.01 square inchbecause 100 of them fill a square inch

A row of 10 tiny squares is 10/100 sq. in.= .10 sq. in. = .1 sq. in.

Zoom in

2.3 inches × 3.6 inches = ? 3x2=6 sq. in.

3 sets of 3 rows of 10 tiny squares= 3 x .3= .9 sq. in.

2 sets of 6 rows of 10 tiny squares= 2 x .6= 1.2 sq. in.

3 rows of 6 tiny squares= .3 x .6= .18 sq. in.

6+.9+1.2+.18 = 8.28

Base ten blocks on the square meter mat

Materials, for each group of 8: A square on a table or the

floor measuring a square meter, or a Square Meter Mat

Lots of base 10 blocks Meter sticks or metric tape

measures

Base ten blocks on the square meter mat

Use two meter sticks or tape measures to lay out a square meter on the mat or the floor. Start to fill it in with flats.

How many flats will it take?

What decimal fraction of the square meter does 1 flat cover?

.2 x .3

Is .2 x .3 = .6? Check by doing .2x.3 with base 10

blocks: Find .2 m and .3 m on meter sticks. Lay out a rectangle .2 meter by .3 meter Fill it in with base 10 blocks, using the

biggest and fewest possible pieces. What decimal fraction of a square meter

does the rectangle cover?

Do some computations on the mat Do several 2-digit multiplications. Use

the biggest and fewest possible pieces to fill in the rectangle.

The product is the area, as a decimal fraction of the square meter.

You will also need to know: What fraction of a square meter does the

“long” cover? How much does the small cube cover?

Do these, or make up your own

a. 0.3 x 0.14b. 0.23 x 0.11c. 0.34 x 0.21d. 0.17 x 0.13e. 0.03 x 0.06f. 0.25 x 0.44

Multiplication Makes Bigger? Students who have experience only with

whole numbers often think that multiplication always makes a bigger number than either factor.

Discuss.

Dividing decimals by whole numbers

We have 4.6 kg of strawberries to share between 4 people. How much does each get?

Note: division as sharing. Note: about how much is 4.6 kg of

strawberries? Use base 10 blocks to “deal out” the

weights of berries into 5 piles. (Try it.) Check with a calculator. What does the

calculator answer mean?

Challenge

How can you get the whole number quotient and remainder using ONLY a 4-function calculator (no writing)?

Example: 26 ÷ 8 = 3 r. 2. Calculator says 3.25. 168 ÷ 35 90 ÷ 7 1234 ÷ 47 9876543 ÷ 4321

Dividing decimals

How many 100’s are in 10,000? What operation is this question asking

about? Example: “How many 3’s are in 21?” can

be interpreted as division: 21÷3. Can also be interpreted as measurement, as in how many centimeters are in the length of this pencil?

So question is asking 10,000÷100

Better yet, count place values.

From 100 to 10,000, go up 2 rows

That is, multiply by 10 two times

So there are 10x10 = 100 hundreds in 10,000

More mental math

Do mentally (using the Powers of Ten poster?), then express as a division.a. How many tenths are in 10?b. How many thousandths are in 0.1?c. How many .2’s are in 140?d. How many .002’s are in .14?

Use the poster:a. .01 x .001b. 100 x .0001c. .03 x 4000

Dividing decimals: fraction approach

Dividing can be expressed as a fraction: 2 ÷ 3 = 2/3

In general, you can move the decimal point the same number of places for both divisor and dividend and get the same quotient.

Decimals on the number line A number line in centimeters Show me. How long is

1 centimeter? 10 centimeters? 100 centimeters (= 1 meter)? 1000 centimeters? 10,000 centimeters? 100,000 centimeters?

The moral of the story

Repeatedly multiplying by 10 makes sizes grow VERY fast!

It’s about multiplying, not adding

Number lines and place value? The places are arranged like counting on

a number line; digits are equally spaced. Suggests adding.

But the values are not equally spaced: multiplication.

Zooming out and in on the number line

Spacing of the place values 1, 10, 100, … looks the same when you zoom out.

Zooming out and in on the number line

Spacing of the place values .01, .1, 1, 10, 100, … looks the same when you zoom in.

References for showing powers of 10

Google maps, zoom in or out. Math software such as GeoGebra; zoom

in or outhttp://www.geogebra.org

Powers of Ten video:http://www.eamesoffice.com/the-work/powers-of-ten/

Thank you for your attention! Susan Addington Math Dept., CSUSB saddingt@csusb.edu Personal website:

http://www.quadrivium.infoAppletsFuture: the Measuring the World Store