CMP 2 Grade Six Operations with Fractions Glenda

CMP 2: Grade Six Operations with Fractions Glenda Lappan Milwaukee April 2005

Fraction Units • Bits and Pieces III

Bits II (fractions) 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 c. The 1/2 mark on the number line above?

d. 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, c. 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 d. 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

Estimating Sums and Differences 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. .

Benchmark fractions: 0 1/4 1/2 3/4 1 11/4 11/2 13/4 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 2 + 3 = 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!