More, Less, or Equal to One Whole Give students a collection of fractional parts (all the same type) and indicate the kind of fractional part they have. Parts can be drawn on a worksheet or physical models can be placed in plastic baggies with an identifying card. The task is to decide if the collection is less than one whole, equal to one whole, or more than one whole. Students must draw pictures and/or use numbers to explain their answer. They can also tell how close the set is to a complete whole. Several collections constitute a reasonable task. Example: If done with fraction strips, the collection might have 7 light green strips with a caption that reads “these are eighths.” Students now decide whether the collection is less, equal to, or more than one whole.

Mixed-Number Names 2 5 Give students a mixed number such as 3 . Their task is to find a single fraction that names the same amount. They may use any familiar materials or make drawings, but they must be able to give an explanation for their result. Similarly, have students start with a fraction greater than 1, such as and have them determine the mixed number and provide a justification for their result. 17 4

Zero, One-Half, or One On the board or overhead, write a collection of 8 to 10 fractions. A few should be greater than 1 ( or ), with others ranging from 0 to 1. Let students sort the fractions into three groups: those close to 0, close to ½ and close to 1. For those close to ½, have them decide if the fraction is more or less than ½. The difficulty of this task largely depends on the fractions. The first time you try this, use fractions such as , , or that are very close to the three benchmarks. On subsequent days, use fractions with most of the denominators less than 20. You might include some fractions such as ¾ or that are exactly in between the benchmarks. As usual, require explanations for each fraction. 7 6 11 10 1 20 53 100 9 10 2 8

Close Fractions Have students name a fraction that is close to 1 but not more than 1. Next, have them name another fraction that is even closer than that. For the second response, they have to explain why they believe the fraction is closer to 1 than the previous fraction. Continue for several fractions in the same manner, each one being closer to 1 than the previous fraction. The first several times you try this activity, let the students use models to help with their thinking. Later, see how well their explanations work when they cannot use models or drawings. Focus discussions on the relative size of fractional parts. Similarly, try close to 0 or close to ½ (either under or over).

About How Much? Draw a picture like one of those seen below (or prepare some ahead of time for the overhead). Have each student write down a fraction that he or she thinks is a good estimate of the amount shown (or the indicated mark on the number line). Listen without judgement to the ideas of several students and discuss with them why any particular estimate might be a good one. There is no single correct answer, but estimates should be “in the ballpark.” If children have difficulty coming up with an estimate, ask if they think the amount is closer to 0, ½, or 1.

Examples for About How Much ? 1 1 ? 2

Ordering Unit Fractions 1 3 1 8 1 5 List a set of unit fractions such as , , , and . Ask children to put the fractions in order from least to most. Challenge children to defend the way they ordered the fractions. The first few times you do this activity, have them explain their ideas by using models. 1 10

Choose, Explain, Test Present two or three pairs of fractions to students. The students’ task is to decide which fraction is greater (choose), to explain why they think this is so (explain), and then to test their choice using any model they wish to use. They should write a description of how they made their test and whether or not it agreed with their choice. If their choice was incorrect, they should try to say what they would change in their thinking. In the student explanations, rule out drawing as an option. Explain that it is difficult to draw fraction pictures accurately and, for this activity, pictures may cause them to make mistakes.

Line ‘Em Up Select four or five fractions for students to put in order from least to most. Have them indicate approximately where each fraction belongs on a number line labeled only with the points 0, ½, and 1. Students should include a description of how they decided on the order for the fractions. To place the fractions on the number line, students must also make estimates of fraction size in addition to simply ordering the fractions.

First Estimates Tell students that they are going to estimate a sum or difference of two fractions. They are to decide only if the exact answer is more or less than 1. On the overhead projector show, for no more than about 10 seconds, a fraction addition or subtraction problem involving two proper fractions. Keep all denominators to 12 or less. Students write down on paper their choice of more or less than one. Do several problems in a row. Then return to each problem and discuss how students decided on their estimate. Variations on pg 268, Vol. 1 Teaching Student Centered Mathematics, by John A. Van de Walle.

Dot Paper Equivalencies Create a “worksheet” using a portion of isometric or rectangular dot grid paper. On the grid, draw the outline of a region and designate it as one whole. Draw and lightly shade a part of the region within the whole. The task is to use different parts of the whole determined by the grid to find names for the part. Students should draw a picture of the unit fractional part that they use for each fraction name. The larger the size of the whole, the more names the activity will generate.

Dot Paper Equivalencies Example The shaded region on the grid is: 1/3 because 2/6 because = 1/3 = 1/6 4/12 because = 1/12

Group the Counters, Find the Names Have students set out a certain number of counters in two colors (Example: 16 red, 8 blue). The 24 counters make up the whole. The task is to group the counters into different fractional parts of the whole and use the parts to create fraction names for the red and blue counters. You might want to suggest arrays, or allow students to arrange them in any way they wish. Students should record their different groupings and explain how they found the fraction names. They can simply use X’s and O’s for the counters.

Group the Counters, Find the Names Examples: Make 2 rows of 4 with the 8 red counters. 2/6 And 4 more rows of 4 makes 24. 4/6 8 groups of blue. 4 groups of red. 8/12 4/12

Slicing Squares Give students a worksheet with four squares, each approximately 3cm on a side. Have them shade in the same fraction for each square using vertical dividing lines. Next, tell students to slice each square into an equal number of horizontal slices. Each square is sliced with a different number of slices, using anywhere from one to eight slices. For each sliced square, they should write an equation showing the equivalent fraction. Students should examine their four equations and the drawings. Challenge them to discover any patterns in what they have done. Repeat often with four more squares and a different fraction.

Slicing Squares Examples The fraction is ¾. First draw vertical lines to make fourths in the square (black lines). Shade in 3 of the 4 to represent ¾. Slice an equal number of horizontal slices (blue). For each sliced square, students write an equation showing equivalent fractions. Look for patterns. How can this be explained? ¾ = 9/12 3 x 3 = 9 ¾ = 12/16 3 x 4 = 12 4 x 3 =12 4 x 4 = 16 ¾ = 6/8 3 x 2 = 6 ¾ = 15/20 3 x 5 = 15 4 x 2 = 8 4 x 5 = 20

Addition and Subtraction with Fractions Paul and his brother were each eating the same kind of candy bar. Paul had ¾ of his candy bar. His brother had ⅞ of a candy bar. How much candy did the two boys have together? How would you solve this if you didn’t know the algorithm or anything about common denominators? Build a model of each and see what you can do with it.

Addition and Subtraction with Fractions Susie and Wendy ordered two identical sized pizzas. Susie ate of a pizza and Wendy ate ½ of a pizza. How much pizza did they eat together? 5 6 Use Fraction Strips to solve this problem.

Multiplication with Fractions There are 15 cars in Connor’s Toy collection. Two-thirds of the cars are red. How many red cars does Connor have? Finding the fractional part of a whole number is not much different than finding the fractional part of a whole. In this problem, the group of 15 cars is the whole. Alan’s cookie problem is the same as many sharing problems used in building understanding of many fraction concepts and their relationship to division. Cookies are used so they can be subdivided.

Multiplication with Fractions Alan has 11 cookies. He wants to share them with his three friends. How many cookies will Alan and his three friends get?

Multiplication with Fractions 2 3 Peter filled 5 glasses with liter of soda. How much soda did Peter use? The first factor being a whole number is also important for students to be exposed to.

Multiplication with Fractions 1 3 You have ¾ of a pizza left. If you give of the leftover pizza to your brother, how much of the whole pizza will your brother get? Use this and the next two problems to expand on the ideas already presented.

Multiplication with Fractions 1 10 9 10 Someone ate of the cake, leaving only . If you eat of the cake that is left, how much of a whole cake will you have eaten? 2 3

Multiplication with Fractions Elizabeth used 2½ tubes of blue paint to paint the sky in her picture. Each tube holds ounce of paint. How many ounces of paint did Elizabeth use? 4 5

Multiplication with Fractions 2 3 Alex had of the lawn left to cut. After lunch he cut ¾ of the grass he had left. How much of the whole lawn did Alex cut after lunch? When the pieces must be subdivided into smaller unit parts, the problems become more challenging. In this problem, it is necessary to find fourths of 2 things, the 2 thirds of the grass left to cut.

Multiplication with Fractions The zookeeper had a huge bottle of the animals’ favorite liquid treat, Zoo-Cola. The monkey drank of the bottle. The zebra drank of what was left. How much of the bottle of Zoo-Cola did the zebra drink? 1 5 2 3 In this problem, you need to find thirds of four things, the 4 fifths of the cola left in the bottle. The concepts of the top number counting and the bottom number naming what is counted play an important role.