Mathematical Emphasis
Investigation 1—Parts of Squares, Halves, fourths, and Eighths

• Understanding that equal fractions of a whole have the same area
• Understanding that equal parts of shapes are not necessarily congruent—that is, they may have different shapes
• Understanding that cutting and pasting shapes conserves their area
• Becoming familiar with relationships among halves, fourths, and eighths

Investigation 2—Parts of Rectangles: Thirds, Sixths, and Twelfths

• Knowing that equal fractions of different-sized wholes will be different in area
• Becoming familiar with relationships among thirds, sixths, and twelfths
• Using different combinations to make a whole
• Comparing fractions that have “one piece missing”
• Working with fractions that have numerators larger than one

Investigation 3—Ordering fractions

• Using both numerical reasoning and area to order fractions
• Using the size of the numerator to compare fractions that have the same denominator
• Using the size of the denominator to compare fractions that have the same numerator
• Comparing fractions greater than 1 with fractions less than or equal to 1
• Identifying equivalent fractions

Tips For Helping At Home
Questions To Ask:

• What is the problem about? Tell me in your own words.
• What did you do in class to get started?
• Can you solve a simpler version of the problem?
• What have you already tried? What steps did you take?
• Did you show all of your work?
• Does the answer make sense?
• How do you know your answer is correct?

Activities at Home

• Look for opportunities to have your child help you divide things up in different way—cutting a cake, slicing the pizza.
• Play Investigations games sent home in this unit: “Fraction Fish”
• Help your child come up with unusual ways to divide things and have him/her prove that everyone gets the same amount.

Vocabulary Terms

Area

The size of a surface measured in square units

Denominator

The number of parts the whole is divided into; the bottom number of a fraction

Equivalent

Having the same value or amount

Fraction

Any part of a group, number, or whole

Numerator

The number of parts of the whole; the number above the line of a fraction

Perimeter

The distance around the outside edge of a figure

Mathematics Vocabulary Web site

Mathematics Strategy—Students' Work on Fourths and Eights
Students are likely to quickly see some of the common ways of dividing a square into fourths and then into eighths, but they will have to think harder about more irregular ways. See figures 1 and 2 below.

One interesting way to generate fourths is to start with a simple division into fourths and to cut a small piece from each fourth to add to the next fourth as in Figure 3. As long as you do the same thing to all four fourths, the resulting figure will also be divided into equal fourths.

When working with eighths, students will likely first come up with a picture like Figure 2 above. Others may then modify it to something like the examples below. More unusual eighths may emerge more slowly, as these require a deeper understanding.

Source: Investigations in Number, Data, and Space: Different Shapes, Equal Pieces. Dale Seymour, 1998. (Page 16)

Mathematics Game—Fraction Challenge

Materials

One deck of fraction cards (any kind of fraction cards can be used for this game

Players: 2 or more

Playing the Game

1. Divide the deck into equal-sized piles, one for each player. All cards stay upside down.



2. Each player turns over the top card in his or her pile. The player with the largest fraction wins. The winner gets all of the other players’ cards and adds them to the bottom of his or her own cards.



3. If two of the cards are equivalent, those two players turn over a second card. The larger fraction wins all of the cards from that round.



4. The game can be won in two ways. Play can continue until one player holds all of the cards and is the winner. A second method is to play for a specific time period and the player with the most cards wins.

Get to Fraction Cards (for printing)