CMP2: Grade Six Operations with Fractions Glenda Lappan Milwaukee April 2005

Fraction Units Bits and Pieces I Bits and Pieces II Bits and Pieces III

Bits II (fractions) Bits II Bits III (Decimals)

Fractions The whole or unit Partitioning/Re-partitioning Naming parts Equivalence— scaling/ ratio/ relative frequency Interpretations –Measures –Indicated division –Operator (stretcher or shrinker —scale factor) –Number (location on a number line) –Ratio

Comparing fraction strips; What is equivalent?

Moving from fraction strips to number lines.

a. On the number line below, carefully label marks that show where 1/3 and 2/3 are located. b.What is the distance from the 1/3 mark to The 1/2 mark on the number line above?

a. What is the distance between the marks for 3/5 and 7/10 on the number line below? b.Locate marks for 1/10, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10, and 10/10. Which of the marks can also be labeled in fifths? c.Find all fractions with denominators smaller than 50 that are equivalent to 10/15

Big Ideas Equivalence Operations Algorithms Solving problems

Equivalence Generating equivalent fractions— ratios/relative frequencies/scaling Fraction to equivalent decimal Decimals to equivalent fractions Expressions Mathematical sentences

Operations Meaning/ What question(s) does the operation answer? What do the computed answer and remainder tell you? Estimating results –Addition –Subtraction –Multiplication –Division

Playing Getting Close

Stop and think about the size of the answer to a problem before you do an actual computation. You can use your knowledge of benchmarking with fractions to know that 3/ 7 + 9/ 20 is greater than a half, but less than one. This is because both 3/ 7 and 9/ 20 are less than, but close to 1/ 2.. Estimating Sums and Differences

Benchmark fractions: 0 1/41/23/ /4 1 1/2 1 3/4 2 Which benchmark is 5/8 nearest? Here is one way to reason: Five-eighths is larger than 1/2, because it is larger than 4/8 Five-eighths is smaller than 3/4, because it is smaller than 6/8 In fact, 5/8 is exactly halfway between 1/2 and 3/4. Which benchmark is 0.58 nearest? Since 0.50 is equal to 1/2, 0.58 is larger than 1/2. We also know that 0.58 is less than 0.75 or 3/4. So we can say that 0.58 is between 1/2 and 3/4, but closer to 1/2.

Solving Problems Deciding which operation(s) to use and why Computing Interpreting computed answers back in original problem

How are operations related? Inverse operations –+ and - –X and ÷ Fact families Finding missing addends and factors Relationship between + and X Relationship between - and ÷

Relating computation to what students already know

Fact Families = 5 has these two related subtraction sentences: 5 – 2 = 3 and 5 – 3 = 2 This set of sentences is called a fact family.

Models for Multiplication

A pan of brownies costs $24 dollars. You can buy any fractional part of a pan of brownies. You pay that fraction of $24. For example, half a pan costs 1/2 of $24. A. Mr. Sims asked to buy half a pan that was 2/3 full. What fraction of a whole pan did Mr. Sims buy and what did he pay? B. Aunt Serena bought 3/4 of another pan that was half full. What fraction of a whole pan did she buy and how much did she pay?

What Motivates Students to Engage with Mathematical Problems?

Interesting challenges!