Is your curriculum optimized for Common Core math?

How is the transition to Common Core working for your class? Does it seem like you have more to teach than before? If so, here are some critical questions to consider when adjusting a unit of instruction from a prior year or from older curriculum materials.

1) How close of a match are the content goals of the unit to the goals of Common Core for your grade?
For quick reference, see the goals list for your grade level at www.mathpaths.com. For more depth, read the actual wording of the standards using the CCSS links on that same page. Make a list and mark the unit goals as required, not required, or uncertain.

2) Which content goals in the unit should be considered as review or as preparation for the next year? 
The color coding and domain icons in the list at www.mathpaths.com will help you find related content in the prior or next grade. If the goal is review, you might be able to give it less emphasis. If the goal is not needed until the next year, considering skipping it! It is much more important for students to learn all goals for this year than to preview goals for the next year. Instead of uncertain, mark that goal as not required. When you skip content, remember to remove related questions from tests or assessment materials.

3) Does the unit include appropriate emphasis on the Standards for Mathematical Practices (MP)?
Mathematical practices are the important habits or processes needed to effectively solve problems including word problems and non-routine problems. These habits are often called problem-solving skills. For Common Core, it is important that problem-solving pages or activities relate to content standards of the prior year or current year. If a publisher's correlation mentions only an MP standard for a lesson, you may be able to skip the lesson. You can increase emphasis on MP standards during other lessons that develop required content.

4) Does Common Core require content that is missing or not sufficiently covered in the unit from a prior year or older materials?
If publishers simply updated the curriculum with "gap" lessons, there may be minimal content for some standards. And, the "gap" content might not be assessed. You may need to supplement the unit with free or low cost activities from TeacherPayTeachers.com, TeachersNotebook.com, or other sources.

In the past, most curriculum materials included 100 to 150 content goals/objectives per year. With CCSS, there are only 50-60 goals/objectives (based on the goals at www.mathpaths.com). Compared to prior years, the curriculum should include fewer topics in more depth! Good luck with Common Core!

Welcome back to school! The summer has been busy!

Can you believe that August is over? The summer has gone by too quickly. I've been busy updating checklists and making other resources for Common Core!
 
New Icons ~ Letter-shaped domain icons have been added to all checklists and posters to help students and teachers identify domains (strands) quickly, even if the signs are printed in black/white.

 

FREE Goals Leaflets using “I Can” Statements ~  There are now free leaflets of goals for each grade from K-8, numbered to match the goals on my poster checklists. These require just one sheet of paper, printed on both sides and folded to make a 4-page leaflet. You can download these from my website, www.mathpaths.com.

New Class Goal Signs ~ The 50-60 goals on each grade’s checklist are now available as individual goal signs illustrated with the domain icons, to be printed or displayed on a screen.

New and Expanded Products for the Standards for Mathematical Practice ~  In 2012, I developed “Be a Star” posters to emphasize practice standards in grades 3 to 8. Over the summer, I wrote a simpler K-2 version of "Be a Star" posters. Versions for 3-8 and K-2 include unique star-shaped icons for each practice standard. Both versions also include a mini book.

Have a great school year!

Angie

Choosing K-8 textbook lessons important for Common Core

The Common Core standards do not tell you how to teach, but they do tell you what to teach. If you are confused about what topics are important, this post can help you.

First of all, let me warn you not to depend on a correlation from the publisher! If the publisher updated textbooks instead of designing a brand new series, be especially wary. Because each change to a textbook affects supplements as well, a publisher trying to meet new standards at minimal cost may keep unneeded lessons and just add "gap" lessons. Many parts of the publisher's correlation may be "stretched." For example, the publisher may correlate a lesson to mathematical practices (problem solving) even when most of the lesson content is not required by CCSS until a later grade. Perhaps only a small part of a lesson is required for CCSS.

The math standards are broken down by domain (strand), with one to four clusters of content for each domain. In K-8, each grade has about 10 clusters in all. At the standards website these clusters are described with paragraphs. My MathPaths website shows lists of lesson goals or objectives based on CCSS cluster descriptions.

Here is a basic approach for evaluating lessons.

1. Look at the lesson title and identify the domain. On my website, domains are color coded to help you see related content across grades.
2. For that domain at your grade level, look for the lesson title and/or content in the cluster descriptions and goals. (The goals that I have written are often similar to common lesson objectives.)
3. If you see a close match, you should use the lesson. If not, you might be able to skip the lesson!
4. Look at the related content at the prior grade and the next grade. If the content is required for two successive grades, pay close attention to the difference between requirements and decide what adjustments to the lesson may be needed.

Because Common Core requires fewer topics per grade, you may find that you can skip a lot of textbook lessons. This should allow time for extra activities toward the goal of getting all students to master the core content. Advanced students may have time to work independently on previously skipped lessons.

Why Use Visual Models before Algorithms?

In Common Core, many standards refer to using visual models such as arrays or area models. For example, students in Grade 4 are expected to use place value, arrays, or area models to multiply 2-digit by 2-digit numbers. In Grade 4, students are not required to know the standard multiplication algorithm. This algorithm is delayed until Grade 5. Here is a visual model for multiplying 17 by 14. Notice how the area grid has four parts. By finding the area of each part and then adding, students can find the product without learning a new algorithm.

I feel that requiring students to understand visual models before using algorithms is extremely important for three reasons.

• Many students don't slow down to think about problems. Once they know a shortcut, they just try to get answers quickly. Visual models help students learn that answers need to make sense!
• The visual model is a bridge to understanding the algorithm. With 2-digit by 2-digit multiplication, the model helps students see that both tens and ones need to be multiplied and then added.
• Students often have difficulty with word problems. Drawing visual models for problems helps students make the connection from the word problem to the related operation.

Although drawing and using models requires time and effort, the result is that students will develop a deeper understanding of important concepts.

What's New with Problem Solving in Common Core?

In addition to content standards, Common Core includes 8 practice standards that are the same for all grade levels. These standards are designed to improve students' problem-solving abilities. The first five practice standards are very similar to the process standards that NCTM has advocated for years. However, there are a few notable differences in Mathematical Practices 6, 7, and 8. I feel that the wording of these is advanced for elementary grades, but the ideas are critical to helping students become proficient with solving problems.

MP6: Attend to precision.

Precision involves using appropriate vocabulary, measurement units, graph labels, and symbols. It also includes adjusting answers to be appropriate for the context of the problem. For example, if the answer to an area or volume question has more nonzero digits than the given information, digits that don't make sense should be rounded. In summary, students need to "pay attention to details" when writing or showing answers to problems. For my students, I extend this to include neatness in written work.

MP7: Look for and make use of structure.

Proficient students can discern patterns and structure in problems. In both geometric and numeric problems, students should be encouraged to break apart or combine parts as they look for patterns. It may also be helpful to sort or rearrange information. An easier summary statement for this standard is to "break apart problems."

MP8: Look for and express regularity in repeated reasoning.

With elementary students, I call this the "look for shortcuts" standard. Students should notice similarities between problems and look for general methods and shortcuts. In the past, some math teachers have expected students to show every step in calculations or solving equations. But to be proficient in math, it is important to recognize and apply shortcuts. For example, students in Grades 4-8 may be able to solve an equation such as 500 + x = 560 by mentally thinking of the missing addend. Because of the general relationship between subtraction and finding a missing addend, students should be encouraged to solve this mentally rather than writing a step of subtracting 500 from each side. The use of shortcuts should be rewarded, not penalized.

Overall, the goal of the Standards for Mathematical Practice is to help students learn habits and processes for solving problems. By encouraging students to pay attention to details, break apart problems, and look for shortcuts, you will help them become better problem solvers.

I have developed a set of 12 posters called "Be a Star" that have checklists and an icon for each practice standard written in language that is easier for elementary students to understand. You can find these on the following site:
http://www.mathpaths.com/practice.html

Common Core Math K-8 Makes Sense!

A lot of teachers are anxious about the transition to Common Core. Don't be. Here are three reasons that Common Core makes sense for K-8 math.

• Fewer Topics per Grade
  There are only about half as many new topics per year, so you'll have more time to spend on each topic. Most textbooks in recent years often "covered" 120 to 160 objectives per grade, while Common Core can be broken down into about 60 to 70 objectives per grade. One reason that textbooks included content "a mile wide and an inch" deep is that standards from state to state were inconsistent; publishers shoved in extra content to match checklists for curriculum committees in various states. Textbooks included a lot of duplication from one grade to the next. In CCSS, duplication is minimal. Review of the prior year will occur when extending the concepts and in the context of problem solving.
• More Emphasis on Developing Understanding before Introducing Algorithms
  As a tutor, I work with many students who apply algorithms without regard to meaning. For example, when asked to find half of 18, one student wrote 1/2 x 18/1 = 18/2 = 9. This is a clue that the student does not really understand "half" and was taught the multiplication algorithm for fractions too soon. The Common Core standards often require computation "using place value" or "using models" the year before the algorithm is introduced. As students use models and place value, they develop a deeper understanding of the concepts. Then, when algorithms are introduced, students will be able to understand the algorithms and apply them more appropriately.
• More Emphasis on Fluency
  Studies have shown that students who are more fluent with basic math will be better problem solvers. (Source: http://www.scholastic.com/teachers/article/math-fluency.) With Common Core, the standards are very clear about fluency requirements for various basic facts and operations. And, since teachers have fewer topics to "cover" in each grade, extra time can be devoted to activities and games to develop fluency for the grade-appropriate topics. This will result in more students having fluency with the same basic skills, so the class can more easily tackle problems requiring higher-order thinking skills.

You may wonder if the Common Core requires "harder" math than before. Generally for Grades K-6, the same content is included but topics may appear in an earlier or later grade or are broken down more. For example, there is a specific requirement for dividing a whole number by a unit fraction in Grade 5, while most division of fractions is the next year. Grades 7-8 do seem somewhat "harder" in that the standards require more algebra and statistics than before. However, there is the expectation that if the curriculum in Grades K-6 has more depth and focus, students will be ready for Grades 7-8!

In another post, I'll talk about the important role of the new Standards for Mathematical Practice.

Generalizing Area as "Average" Length x Width

Why are there so many different formulas for area? Many students become dependent on applying formulas, but they forget the formulas soon after the chapter test.

In tutoring, I emphasize that area is ALWAYS related to length by width. I encourage students to think about the middle or average length of a shape. Since the area is a measurement of square units, segments representing length and width must always be perpendicular to each other. For consistency, I have students consider the length as the horizontal measurement, and the width as the vertical measurement (which can also be called the height).

Here are some examples.

Consider parallelograms and trapezoids, as shown by the blue shapes above. If you draw a line segment across a parallogram, parallel to a base, that segment will always be the same length as the base. So, it can be considered the "average" length. For a trapezoid, the middle segment is the same as half of the sum of the two bases. For triangles, as shown in purple above, the middle length is half of the base.

I often have students identify the length and width of a rectangle by counting the squares on graph paper. Then I  extend this activity to identifying and labeling the middle segment of a parallelogram, trapezoid, or triangle. In this way, students are better able to understand the concept of area of these types of shapes and may not need to memorize individual formulas.

In Common Core, students in Grade 2 are expected to partition rectangles into square units (2.G.2). In Grade 3, students count squares to find the area, and relate area to multiplication (3.MD.6 and 3.MD.7). In Grade 4, students calculate areas of rectangles (4.MD.3). In Grade 6, students find area of other polygons (6.G.1 and 6.G.4).

Please write a comment if you have your own special methods for teaching area.

Steps for Solving Equations

In CCSS, the description of Standard 8 for Mathematical Practice says, "Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts."

This week, I was tutoring an 8th grade student who was very bogged down when solving equations. Her teacher, like many others, was requiring students to "show all of the work." Here is an example of work that the student was showing to solve the equation 7x – 17 = 11. Notice that her first step was to write +17 under each side and draw lines to add. My concern is that this step detracts from key idea of equation solving, which is to write equivalent equations that become simpler and simpler until the solution is clear.

In most textbooks, the second line above would be shown as an equation:
7x – 17 + 17 = 11 + 17
This is OK since it is an equivalent equation, but the goal should be for students to do more steps mentally, as follows (in black text):

When I encourage students to use shortcuts and show only key steps, they usually respond positively. They are glad to try to do more of the work mentally, and it is easier for them to see the progression to simpler and simpler equivalent equations. As a result, students become more confident and their homework papers look neater!

Another Round of Pi

Some of you may have discussed pi with students on March 14, often called Pi Day. Are you ready for another look at that special number that is very close to 3.14?

Many students are confused by the formulas for circumference and area of a circle. In CCSS, circle relationships and formulas are to be taught in Grade 7, standard 7.G.4. To help my students with circumference and area, I've developed a visual approach.

First consider the circumference or distance around a circle. I ask leading questions to help students estimate the circumference as a multiple of the length of the diameter, based on these diagrams.

For area, I help students relate the area of a circle to the square of the radius.


I help students see that when the diameter of a circle is 1, the circumference is pi! And, if the radius is 1, the area is pi! If students visualize the relationships between parts of circle, they are more likely to develop an understanding of the formulas.

How Dot Arrays on Grids Cause Confusion

A few days ago during math tutoring, a 6th grader was having trouble counting the length and width of a rectangle on a grid. When shown a rectangle 4 units wide and 2 units long, she said the area was 15 square units! The error was caused by counting the intersection points instead of the spaces. Consider the following figures.

Some curriculum materials include practice with making dot arrays on a grid for multiplication. Students who make these arrays may then be confused about finding the length of a segment on the grid. In Common Core, students are expected to draw and count arrays of squares. This is a better approach because students are less confused and it leads perfectly into the concept of area.

Simplified Metric Conversions with Common Core

Recently, one of my 7th grade students needed help with metric conversions. The teacher had given her a "staircase" diagram. From upper left to lower right, the steps were marked kilo, hecto, deca, (base unit m/L/g), deci, centi, and milli. The student was told that to convert by counting steps and then moving the decimal point. For example, meters to kilometers is three steps to the left, so move the decimal point three places to the left.

Even though using this type of diagram can lead to correct answers on homework, there are two reasons why this method troubles me.

• Students are dependent on the diagram. Most students will not remember how to draw and label it if they need to convert metric units a week, month, or year later.
• Many of the metric prefixes on the diagram are rarely used! In fact, the only prefixes needed for high school courses and tests such as ACT and SAT are kilo, centi, and milli.

I'm very glad that for CCSS, students are expected to convert measurements using two-column tables. In Grade 4, students convert from larger units to smaller units. In Grade 5, students convert in either direction. The units mentioned in the standards for Grade 4 are kilometer, meter, centimeter, kilogram, gram, liter, and milliliter.

To convert a measurement, first set up a table with headings for the two units. Under the correct headings, record the relationship and also the measurement to be converted. Here are two examples. The table at the left is set up for converting 12 liters to milliliters (grade 4). The table at the right is set up for converting 15 millimeters to centimeters (grade 5).

From tables like these, it is easy for students to see whether to multiply or divide to convert the measurements! Writing relationships in tables is also excellent preparation for variables and functions in higher grades.

Relating Decimals to Fractions (Grades 4-5)

Many students who come to me for math tutoring are very weak with fraction and decimal concepts. Here are a couple of sample responses from 5th grade students this week.

Q: How can you write 0.15 as a fraction?
A: Is it one-fifth?

Notice that the student tried to write the fraction using the same digits as the decimal.

Q: Write the number two and fifty-four hundredths.
A: 2.54
Follow-up Q: That's right. Can you also write it as a mixed number?
A: No, I don't know how.

Many students have trouble relating decimals to fractions because of inadequate emphasis on decimals as shortcuts for fractions. In Grade 4 of Common Core, there is a high emphasis on fractions in tenths and hundredths, and on meaning of decimals related to fractions. Here are examples of expectations from Common Core 4.NF.5 and 4.NF.6.

• Express fractions in tenths as hundredths.  (i.e. 2/10 is the same as 20/100)
• Add fractions in tenths and hundredths. (i.e. 3/10 + 4/100 = 34/100)
• Convert between decimals and fractions in tenths or hundredths (i.e. 4/10 = 0.4, 0.78 = 78/100)

It is very important that students have a strong understanding of fractions before tackling decimals or students will simply be manipulating digits. They may be getting the correct answers to decimal questions, yet have no idea what the decimals mean. In my tutoring, I emphasize that every number in decimal form is really just a shortcut for a fraction or mixed number.

With students in grades 4-8, I often show them how to change long decimals to fractions. For example, 0.123456789 is the fraction 123,456,789/1,000,000,000. Each digit to the right of the decimal point corresponds to a zero in the denominator of the fraction.

Tales from Math Tutoring

As a mathematics tutor, I help individual students overcome their misconceptions. Students can be confused in many surprising ways! In this blog, I will be giving examples of interesting errors and comments by my students. Usually, I'll follow up with suggestions based on Common Core State Standards that will help minimize those types of errors and misconceptions. I hope you'll follow my blog and send me your own stories!