January 12, 2011 Monica Hartman. A rug is made of very small knots that each fills one square millimeter. How many knots will the rug makers have to tie.

January 12, 2011Monica Hartman

A rug is made of very small knots that each fills one square millimeter.

How many knots will the rug makers have to tie to make one square meter?

4.NF.7: Compare two decimals to hundredths by reasoning about their size.

4.MD.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems that require expressing measurements given in a larger unit in terms of a smaller unit.

5.NBT.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. (N.ME.05.08)

5.MD.1: Convert like measurement units within a given measurement system (e.g., convert 5cm to .05m), and use multi-step real world problems

Use your fingers, hands, and arms to show: mm cm dm m

square meter square decimeter square centimeter square millimeter

10 x 10 =

Prerequisite Knowledge Prerequisite Knowledge or Use Calculatorsor Use Calculators

100

1,000 10 x 100 =

10 x 1,000 = 10,000

10 x 10,000 = 100,000

100 x 100 = 10,000

100 x 1,000 = 100,000

1,000 x 1,000 = 1,000,000

Cut out a square decimeter on your grid paper.

How should you place the square decimeters to make a square meter?

Solve the problem in your group. Make sure everyone in your group understands the problem and can explain it.

If you are finished ahead of time, solve the problem in another way.

Representing and Solving Representing and Solving the Problemthe Problem

A rug is made of very small knots that each fill one square millimeter.

1 square decimeter

In 1 square centimeter there are 100 knots (10 mm x 10 mm)

How many knots will the rug makers have to tie to make one square centimeter?

In 1 square decimeter there are 100 x 100 knots or 10,000.

A rug is made of very small knots that each fill one square millimeter.

1 square meter

1 square decimeter, with 10,000 knots

How many knots will the rug makers have to tie to make one square meter?

In 1 square meter there are 10,000 x 100 knots or 1,000,000.

How many centimeters are long each side of a square meter?

100 How many millimeters are along each side of the

square meter ? 100 x 10 = 1,000

How many square millimeters are in a square meter?

1,000 x 1,000 = 1,000,000

Another RepresentationAnother Representation

How easy/difficult was the rug problem for you?

What did you do when you got “stuck”? Was that strategy successful or not? Why? Was there something we did in class that

helped you? What do you think was the “big idea” in this

problem? Were you able to learn from or contribute to

the learning of others?

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning

of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Read Lappan article silently in your group.

Get ready to share one idea from the article with the others in your group.

With your group, brainstorm ways that you can help your students develop a good mathematical disposition.

Choose a strategy that you will try for the new year.

List all the ways that they have seen fractions developed in your own learning.

List all the uses for fractions in the real world.

Students have difficulty transferring from shaded circles to fraction of a length.

Children should explore fractions across all three models.

1) The relationship to the whole unit is immediate.

2) Size comparisons are easily made in the ordering of fractions

3) Common denominators are apparent in addition and subtraction

4) Equivalent fractions can be easily generated.

5) All four of the fundamental operations can be represented by folding strips.

Fractions that are compared must be on strips of equal lengths.

Loop the strip, placing the ends together, and crease.

Label each part ½.

Loop the strip one and on-half times, bringing the ends opposite one another, and crease.

Label each part 1/3

Fold two equal parts. Fold each of them in half for a crisp fold. Label each section, fold back and forth.

Loop the ends all the way around and to opposite sides of the loop, where it will be creased.

The ends need to be at the creases, so before creasing, flatten the ring and adjust if necessary.

Fold back and forth (zig-zag fold) to assure that all five sections are of equal length.

Fold the 1/3s in half or fold the 1/2s into thirds.

Fold, crease, and label two more strips to complete the set.

4.NF.1: Explain why a fraction a/b is equivalent to a fraction (n*a)/(n*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.

4.NF.2: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, <, and justify the conclusions, e.g., by using a visual fraction model.

4.NF.3: Understand a fraction a/b with a>1 as a sum of fractions 1/b. a) Understand addition and subtraction of fractions

as joining and separating parts referring to the same whole.

b) Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 or 3/8 = 1/8 + 2/8;

2 1/8 = 1 + 1 +1/8 or 2 1/8 = 8/8 + 8/8 + 1/8

5.NF.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (in general, a/b + c/d = (ad + bc)/bd.

5.NF.2: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7 by observing that 3/7 is less than 1/2.

Ask students to move the cards around to learn how they can be arranged to show decimal numbers.

Have students select the cards that can be used to show the decimal number 0.268.

Ask students to place the cards on top of each other to display the numbering standard form.

Model the same decimal number on the decimal mat.

Which decimal number is greater? 0.6 or 0.28

Have students find the Decimal Seret Code cards for 0.3 and .03 and .003. Using the cards and the < and> signs inequality signs ask them to arrange all possible comparisons of the three decimal numbers. Repeat the activity using the cards for .8, .08 and .008.

Find the Decimal Secret Code cards for 0.3 and 0.03 and 0.003.

Use the > and < signs to write all possible comparisons.

0.3 > 0.03 0.03 < 0.3 0.3 > 0.003 0.003 < 0.3 0.03 > 0.003 0.003 < 0.03

Find the Decimal Secret Code cards for 0.8, 0.08 and 0.008.

Use the > and < signs to write all possible comparisons.

0.8 > 0.08 0.08 < 0.8 0.8 > 0.008 0.008 < 0.8 0.08 > 0.008 0.008 < 0.08

4.OA.3 Solve multistep word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies, including rounding.

Amy has 5 times as many stickers as Ben. Together they have 54 stickers. How many more stickers does Amy have than Ben?

Go to Thinking Blocks to see how they represent this problem.

Continue to explore Thinking Blocks.

Can these ideas be useful without technology?

Design a rich open-ended problem that can be solved in multiple ways.

Give this problem to your students. Use the problem solving structures we use

in class. Allow time for student reflection. Bring samples of students’ work and

reflection next time.