Fraction Sense*

How do we get there?

Decimal DisconnectsAnd, what to do about them…

*which includes decimals, and percent…

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Why Fractions?

• 50% of 8th graders could not order three fractions from least to greatest (NAEP, in NCTM, 2007).

• Less than 30% of 17-year olds correctly represented 0.029 as 29/1000 (Kloosterman, 2010).

• When asked which of two decimals, 0.274 and 0.83 is greater, most 5th and 6th graders chose 0.274 (Rittle-Johnson, Siegler, and Alibali, 2001).

Grade 4.NFA. Extend understanding of fraction equivalence and ordering.

1. Explain why a fraction a/b is equivalent to a fraction (n x a/n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

2. Compare two fractions with different numerators and denominators

C. Understand decimal notation for fractions and compare decimals fractions.

5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators of 10 and 100.

6. Use decimal notation for fractions with denominators 10 or 100.

7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of the comparisons with the symbols <, =, or >, and justify the conclusions (e.g. by using a visual model.

Grade 5.NBT

A. Understand the place value system.

1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of ten.

3. Read, write, and compare decimals to thousandths.

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 x 3 x (1/10) + 9 x (1/100) + 2 x (1/1000).

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results.

4. Use place value understanding to round decimals to any place.

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What’s important here?

• Representations – models, drawings

• Equivalence – very important

• Comparing and Ordering Decimals – linked to equivalence

• Connections – connecting math-wise (e.g. to fractions, ratio, proportion)

• Contexts – tasks in settings (e.g. measurement)

Fennell, Kobett, and Wray, MTMS (2014)

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Think about and do…

• Represent the following using what’s in front of you:

0.2

0.87

1.23

0.999 (good luck)

• What materials do you use for decimals?

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0.7• What happens to the value of the decimal

if a 0 is inserted AFTER the 7?

• What happens to the value of the decimal if a 0 is inserted BEFORE the 7?

• How would you demonstrate that 0.15 > 0.149

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Which is closest to 1?

0.9

1.1

0.91

1.09• How do you know?

Fennell, Kobett, and Wray, MTMS (2014)

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Comparing and Between

• Think of a number between the following – show how you know.1/4 and 1/3- 0.29 and - 0.30

16% and 19%

• Note: finding a fraction that falls between is more challenging (for most) than finding a decimal or percent.

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More on Comparing

• Give the students cards with the numbers 1, 4, 5, and 9 and one with a decimal point on them (or some other random digits 0-9.. Maybe different pairs get different sets?

• Then ask them to create the smallest number they can. The biggest number they can

• A number that is closest to 5. A number that is closest to 50.

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More on Comparing

• Alan tried to make a decimal number as close to 50 as he could (using 1, 4, 5, & 9). He arranged them in this order: 51.49. Jerry thinks he can arrange the same digits to get a number that is even closer to 50. Do you agree or disagree? Explain your thinking.

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Can I have this student…

Take a look…

Mathematics Teaching Cases

Carne Barnett-Clarke, et al

Grade 5.NBT

B. Perform operations with multi-digit whole numbers and with decimals to hundredths.

7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

What do we do about this?

Decimal Contexts

Grade 4.MD

A.Solve problems involving measurement and conversion of measurement from a larger unit to a smaller unit.

Grade 5.MD

A. Convert like measurement units within a given measurement system.

What contexts do you use? Advocate for?

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What about percent?

• Common percents and you?• 6.RP.A.3.c – Find a percent of a quantity

as a rate per 100; solve problems involving finding the whole, given a part and the percent.

• 7.RP.A.3 – Use proportional relationships to solve multistep ratio and percent problems (interest, taxes, etc.).

• Standards and curriculum!

Grade 6 – Ratios and Proportional Relationships

A. Understand ratio concepts and use ratio reasoning to solve problems.1. Concept of ratio

2. Concept of unit rate

3. Ratio and rate reasoning

Grade 7 – Ratios and Proportional RelationshipsA. Analyze proportional relationships and use them to solve real-world

and mathematical problems.1. Compute unit rates associated with ratios of fractions…

2. Recognize and represent proportional relationships between quantities.

3. Use proportional relationships to solve multistep ratio and percent problems.

Coming next:middle school…

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Grade 5• Extending division to 2-digit divisors, integrating decimal fractions

into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations.

Grade 6• Connecting ratio and rate to whole number multiplication and

division and using concepts of ratio and rate to solve problems;

Critical Areas