Overview of the

2011 Massachusetts Curriculum Framework for Mathematics

Incorporating the Common Core State Standards for Mathematics If you are interested in seeing the entire set of Massachusetts Mathematics Standards please click on the link

http://www.doe.mass.edu/frameworks/math/0311.pdf

This presentation is intended to illustrate some of the changes in elementary math as a result of the new standards.

1) Teaching Strategies 2) Addition & Subtraction 3) Multiplication & Division

A major strength of the Common Core is its unity of teaching strategies and teaching tools in all grades. The presentation deliberately highlights this unity.

It all begins with … The 8 Standards for Mathematical Practice These standards are common across all grade levels Kindergarten through Grade 12.

The 8 Standards for Mathematical Practice are listed below. The goal of the new standards is to ensure students are using these skills daily to connect one skill/concept learned to the next.

1. 2. 3. 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

If you are interested in learning more about these standards please click on the link If you are interested in learning more about these standards please click on the link

http://thinkmath.edc.org/index.php/CCSS_Mathematical_Practices

describes how the Standards of Mathematical Practice relate specially to the elementary classroom.

. This website . This website

These standards are best understood by grouping them. Numbers 1 and 6 are the nuts and bolts of mathematics teaching:

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

6. Attend to precision.

Students need to solve problems precisely.

Standards 4 and 5 deal with types of problems used and and how they are solved:

4. Model with mathematics.

5. Use appropriate tools strategically.

Modeling is essentially using real world situations . Tools has a wide meaning, many new “tools” in elementary are explained in this presentation.

Standards 2 and 3 deal with reasoning:

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

Students need to reason about problems, explain their reasoning to others, and understand the reasoning of other people.

Standards 7 and 8 deal with understanding and using the basic structure of our number system:

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

These two standards are used extensively in teaching calculation skills . They are used throughout this presentation.

Standards for Mathematical Teaching Practice

1. 2. 3. 4. 5. 6. 7. 8. Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

The use of ten frames, hundreds charts, number lines, arrays and non traditional algorithms all fall within these standards.

When using these tools strategically and appropriately, we are teaching students to make use of structure and repeated reasoning.

These mathematical practices are linked together by how we teach numbers.…...for example…..make use of structure…… Compose & Decompose Numbers.

5 = 2 + 3 5 = 4 + 1 5 = 10 - 5

This skill is used repeatedly in teaching computation. Over time, students become very agile at composing and decomposing numbers. All of the following are used later in this presentation.

50 = ½ x 100 14 = 2 x 7 7 = 5 + 2 13 = 10 + 3 12 = 10 + 2 12 = 2 x 6 27 = 20 + 7 35 = 30 + 5 6 = ½ x 12 25 = 20 + 5

The skill of subitizing is essential. It is now explicitly taught and another fundamental building block.

• su·bi·tize [soo-bi-tahyz] verb (used without object), -tized, -tiz·ing. -to perceive at a glance the number of items presented, the limit for humans being about seven.

As adults, we all recognize these dot patterns as 5 and 6 without having to count them. This is subitizing and it is the foundational skill to numeracy.

Students should practice subitizing skills with the dot patterns on both number cubes and ten frames. Students should also be able to recognize how many fingers are held up without having to count.

.

For more subitizing activities you can go to: Below are links to activities that will develop subitizing skills.

http://www.edplus.canterbury.ac.nz/literacy_numeracy/maths/numdocuments/dot_card_and_ ten_frame_package2005.pdf

http://illuminations.nctm.org/ActivityDetail.aspx?ID=74 http://illuminations.nctm.org/ActivityDetail.aspx?ID=218 http://illuminations.nctm.org/ActivityDetail.aspx?ID=219 http://illuminations.nctm.org/ActivityDetail.aspx?ID=73

Subitizing with 5 is extremely important….

Connecting subitizing to addition & subtraction

Students should avoid using an un-organized pile of objects to add and subtract….

8 + 7 =

We want to encourage students to use the structure of the ten frame tool to solve problems. It is the frequent use of the structure and tools that will allow students to ultimately become flexible in their thinking.

8 + 7 = 15

8 + 7 = 15

Solving for an unknown using the structure of the ten frame tool. The same ten frame can represent different problems.

8 + =10 10 - 2 =

OA standards based on solving for unknown

In presenting all problems, it is important to phrase and discuss the same problem in different ways, enhancing problem solving and modeling.

Doing calculations…….

with understanding.

Addition & subtraction learned based on place value using tools to find structure & generalize. (Back to standards of teaching practice)

Four

BIG

tools: 1) Ten Frames 2) Hundreds Charts 3) Number Lines 4) Alternate Algorithms

Ten Frames

–

Build on subitizing 5

–

Composing & decomposing

–

Flexible thinking

18 + 17

NBT standards based on place value

18 + 17 = 20 + 15 = 35

NBT standards based on place value

Hundreds Chart

– Look for and make use of structure.

– Look for and express regularity in repeated reasoning.

26 +13

26 +13

26 +13

Number Line……a great tool for relative structural position of numbers.

+ 10 + 2 49 59 61

49 + 12 = 61

Non-Traditional Algorithms

–

Lead to traditional algorithms.

–

Based on use of tools.

–

Use structure of the number system.

An example of the Partial Sums Algorithm: 1 8 + 1 7 2 0 Step 1: Add the tens (10 + 10 = 20) and write the sum.

NBT standards based on place value

An example of the Partial Sums Algorithm: 1 8 + 1 7 2 0 1 5 Step 1: Add the tens (10 + 10 = 20) and write the sum.

Step 2: Add the ones (8 + 7 = 15) and write the sum. NBT standards based on place value

An example of the Partial Sums Algorithm: 1 8 + 1 7 2 0 + 1 5 3 5 Step 1: Add the tens (10 + 10 = 20) and write the sum.

Step 2: Add the ones (8 + 7 = 15) and write the sum. Step 3: Add the partial sums to get 35 NBT standards based on place value

An example of the Partial Sums Algorithm: 1 8 + 1 7 2 0 + 1 5 3 5 Step 1: Add the tens (10 + 10 = 20) and write the sum.

Step 2: Add the ones (8 + 7 = 15) and write the sum. Step 3: Add the partial sums to get 35 Note connection with ten frames NBT standards based on place value

Examples Student Problem Solving 13 – 9 = ____

Student Problem Solving

Student A

13 - 9

I know that 9 plus 4 equals 13.

So 13 minus 9 is 4.

Student Problem Solving

Student B

13 - 9

Instead of 13 minus 9, I added 1 to each of the numbers to make the problem 14 minus 10. I know the answer is 4. So 13 minus 9 is also 4.

Student Problem Solving

Student C

13 - 9

9 is 3 and 6.

13 minus 3 is 10. 10 minus 6 is 4. So 13 minus 9 is also 4.

Multiplication & Division… With Understanding…...

Using Tools, Structure & Repeated Reasoning.

Three BIG Tools: 1) Arrays 2) Area 3) Non-Traditional Algorithms

Arrays • Display the structure of multiplication.

• Display the repeated patterns in multiplication.

2 x 7 is 14…

Arrays mesh with composing and decomposing numbers…to build on previously learned computation skills.

If 2 x 7 is 14… then 4 x 7 =28

Because if you double 2, it will equal 4, So we double 2 x 7 = 14 to 4 x 7 to equal 28!

Here is another area where students will need to be able to apply the skill of composing and decomposing numbers.

Area Model……with composing and decomposing numbers Actually the area model is very similar to arrays. Area should begin as an array of squares……

5 x 7 = 5 7

If you are interested in trying an interactive website with this multiplication tool please click on the link.

http://nlvm.usu.edu/en/nav/frames_asid_192_g_1_t_1.html?from=topic_t_1.html

Decompose the area to create multiplication problems with known answers.

5 x 7 = 5 x 5 + 5 x 2 = 35 5 2 5 5 x 5 = 25 5 x 2 = 10

This model and thinking leads naturally into double digit multiplication... 27 x 30 is 20 + 7 all times 30

27 x 30 = 20 x 30 and 7 x 30 30 20 20 x 30= 600 7 7 x 30 = 210 By this time….you do not need to draw every square

27 x 30 = 20 x 30 and 7 x 30 30 20 20 x 30= 600 7 7 x 30 = 210 The whole area comes from adding the two areas together: 600 + 210 = 810

Continuing this same development... 27 x 35 is 20 + 7 all times the number 30 + 5

27 x 35 20 20 x 30= 600 30 7 7 x 30 = 210 5 20 x 5 = 100 7 x 5 = 35

27 x 35 20 20 x 30= 600 30 7 7 x 30 = 210 600 + 100 + 210 + 35 = 945 5 20 x 5 = 100 7 x 5 = 35

Non-Traditional Algorithms • A tool for bridging to the standard algorithm.

• Build on the area model and its use of structure and repeated reasoning.

An example of the Partial Products Algorithm: X 35 27 35 7 x 5 = 35 NBT standards based on place value Mathematical Practice #8 Look for and express

regularity in repeated reasoning

An example of the Partial Products Algorithm: X 35 27 35 210 7 x 5 7 x 30 = 35 = 210 NBT standards based on place value Mathematical Practice #8 Look for and express

regularity in repeated reasoning

An example of the Partial Products Algorithm: X 35 27 35 210 100 7 x 5 7 x 30 20 x 5 = 35 = 210 = 100 NBT standards based on place value Mathematical Practice #8 Look for and express

regularity in repeated reasoning

An example of the Partial Products Algorithm: X 35 27 35 210 100 600 945 7 x 5 7 x 30 20 x 5 20 x 30 = 35 = 210 = 100 = 600 NBT standards based on place value Mathematical Practice #8 Look for and express

regularity in repeated reasoning

X 35 27 35 210 100 600 945 Note how this Connects to Area. 20 7 x 5 7 x 30 20 x 5 20 x 30 30 20 x 30= 600 = 35 = 210 = 100 = 600 5 20 x 5 = 100 7 7 x 30 = 210 7 x 5 = 35

Examples of Student Problem Solving There are 25 dozen cookies in the bakery. What is the total number of cookies at the bakery?

Student Problem Solving Student 1 25 x12 I broke 12 up into 10 and 2 25 x 10 = 250 25 x 2 = 50 250 +50 = 300

Student Problem Solving Student 2 25 x 12 I broke 25 up into 5 groups of 5 5 x 12 = 60 I have 5 groups of 5 in 25 60 x 5 = 300

Student Problem Solving Student 3 25 x 12 I doubled 25 and cut 12 in half to get 50 x 6 50 x 6 = 300

Division… With Understanding…...

Using Tools with structure & repeated reasoning

Arrays • Display the structure of division.

• Display the repeated patterns in division.

14 stars are arranged into 7 rows. How many are in each row?

14 stars are arranged into 7 rows. How many are in each row?

14 ÷ 7 = 2

The same problem has another wording….

14 stars are arranged into equal rows of 7. How many rows are there?

14 stars are arranged into equal rows of 7. 14 ÷ 7 = 2

Arrays or Area With Composing & Decomposing

35 ÷ 5 means A shape has an area of 35. One side is 5. What is the other side?

35 ÷ 5

5

If you are interested in trying an interactive website with this division tool please click on the link.

http://nlvm.usu.edu/en/nav/frames_asid_193_g_1_t_1.html?from=topic_t_1.html

As with multiplication, this problem solving approach extends to larger numbers, but it is no longer necessary to draw each square.

672 ÷ 3 means a shape has an area of 672 and a side of 3

Decompose the unknown side into know values.

672 ÷ 3 =

3

300 100 372

Continue to decompose the unknown side into know values.

672 ÷ 3 =

3

300 300 72 100 + 100

672 ÷ 3 =

3

300 300 60 12 100 + 100 + 20

When done, add the values to get the length of he missing side. 672 ÷ 3 =

3

12 100 + 100 + 20 + 4 = 224

Non-Traditional Algorithms • A tool for bridging to the standard algorithm.

• Build on the area model and its use of structure and repeated reasoning.

Done out as Partial Quotients Algorithm: 3 672 - 300 372 - 300 72 60 12 12 0 3 x 100 = 300 3 x 100 = 300 3 x 20 = 60 3 x 4 = 12 =224

300 100 + 300 100 + 60 20 + 12 4 3 672 - 300 372 - 300 72 60 12 12 0 3 x 100 = 300 3 x 100 = 300 3 x 20 = 60 3 x 4 = 12 =224

Final Summary Thoughts New Massachusetts Standards • Unified approach to teaching and learning arithmetic & fractions: same tools & same math practices.

• Tools (Ten Frames, Hundreds Charts, Number Lines, Non Standard Algorithms, Arrays, Area Models).

WHY?

• • We want students not to depend upon the tools but to internalize the structure of the mathematical tools they are using. Supporting students with tools and explicitly teaching them to use structure and repeated reasoning will allow students the ability to access and solve more complicated problems.

Final Thought: Ultimate Goal of the New Massachusetts Standards

• Teach students to apply the Standards of Mathematical Practice to think like mathematicians.

Specifically…..

• We want students to be flexible in their thinking so they can solve problems mentally with a deeper understanding of place value.