Tips, Math Practices, and Ideas about Data & Graphs

Topic 19

The standards for this topic are:

Develop understanding of statistical variability.

CCSS.MATH.CONTENT.6.SP.A.1
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
CCSS.MATH.CONTENT.6.SP.A.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
CCSS.MATH.CONTENT.6.SP.A.3
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Summarize and describe distributions.

CCSS.MATH.CONTENT.6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
CCSS.MATH.CONTENT.6.SP.B.5
Summarize numerical data sets in relation to their context, such as by:
• CCSS.MATH.CONTENT.6.SP.B.5.A Reporting the number of observations.
• CCSS.MATH.CONTENT.6.SP.B.5.C Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
• CCSS.MATH.CONTENT.6.SP.B.5.D Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Tips

The performance task at the end of the unit is about students using data and statistics to analyze surveys and sales from a fund-raising project. Why not try this unit as a PBL? Students could look for a real need in the community to fund-raise for.

If you aren’t ready to give that a try, then there are two PBLs that are part of the PBLU classes by BIE:

• In New-Tritional Info, students use rates and proportions to calculate how long they’d have to exercise to burn off different McDonald’s menu items and explore this even further in two project tasks, Have It Your Way and Would You Like Fries with That?”
• In the BizWorld project, students work in teams to form friendship bracelet companies by raising capital (not actual money, but in a scenario).”

Technology Integration Resources

• 6.SP.1 – Recognize a statistical question as one that anticipates variability in …
• 6.SP.2 – Understand that a set of data collected to answer a statistical question …
• 6.SP.3 – Recognize that a measure of center for a numerical data set …
• 6.SP.4 – Display numerical data in plots on a number line, including dot plots, …
• 6.SP.5 – Summarize numerical data sets in relation to their context, such as by:
• 5.a – Reporting the number of observations.
• 5.b – Describing the nature of the attribute under investigation, including …
• 5.c – Giving quantitative measures of center (median and/or mean) and …
• 5.d – Relating the choice of measures of center and variability to the shape …

Math Practices

When you incorporate academic conversations into your math lessons, you are having them attend to precision. Have them focus on using clear and precise language in their discussions with others and in their reasoning. Have them use the appropriate mathematical terms in discussions about data and statistics (including mean, median, and mode).

Incorporate ELA in Math: Perhaps have them research a famous scientist and the data she/he worked with. Then have a mini-reader’s theater where they pretend to be the scientist discussing the data. Have them work in groups, taking turns as the interviewer, interviewee, and keeping track of the academic vocabulary used and used appropriately.

What are some real-world applications of data and statistics that students can discuss?

Tips, Math Practices, and Ideas about Volume and Surface Area

Topic 18- Volume and Surface Area

 18-1 Solid Figures 6.G.4 18-2 Surface Area 6.G.4 18-4 Volume with Fractional Edge Lengths 6.G.2 18-5 Problem Solving: Use Objects and Reasoning 6.G.4

CCSS.MATH.CONTENT.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Technology Integration Resources

There are great resources at CCSS Math, including instructional videos, interactive sites with immediate feedback, performance tasks, etc.

Tips, Math Practices, and Tech Integration

Have you thought about taking the performance task in enVision (or you can use these ones too), and then have students discuss the questions on an online chat such as Today’s Meet, on a blog, in a Moodle forum, or My Big Campus chat? As they are discussing ideas online, you can also include specific questions focusing on math practices such as:

• #3 Construct viable arguments and critique the reasoning of others: Teachers ask clarifying and probing questions and set up a safe environment for students to agree and disagree with one another. Students would then focus on supporting their arguments and listen (read) other arguments and approaches and decide if it is reasonable.
• #1 Make sense of problems and persevere in solving them: Teachers would provide wait time for groups of students to analyze the information and explain the meaning of the problem; then discuss possible solutions. Once they have vetted their ideas, they could compose their thoughts to share online.

In such discourse, the teacher would need to adjust questions. For example, if the teacher realizes that the students don’t quite understand the question being asked, then she/he could ask questions to have them paraphrase the question being asked, have them ask and answer their own questions to clarify what is being asked… And lead them through the problem solving steps through these math practices.

Why online discussion? There are several reasons. First, it is revealing to have students actually narrow down their discussion to the main idea. To do so, they need to know what they are going to say, which means they make a decision about their deductions. Second, some of your less vocal students often times feel more comfortable offering their ideas online. It gives them a platform with equity of opportunities to share.

What other ideas do you have related to this post?

Photo Credit: Guillén Pérez via Compfight

Tips, Math Practices, and Ideas about Perimeter and Area

Topic 17

 17-1 Perimeter ** 6.EE.2.c M06-S4C4-05 17-2 Area of Rectangles and Irregular Figures** 6.EE.2.c M06-S4C4-04, M06-S4C4-05, M06-S5C1-02 17-3 Area of Parallelograms and Triangles** 6.EE.2.c M06-S4C4-04, M06-S4C4-05, M06-S5C1-02 17-4 Step-Up Lesson: Circumference 7.G.4 M06-S4C1-01 17-5 Step-Up Lesson: Area of a Circle 7.G.4 17-6 Problem Solving: Use Objects 6.G.4 18-5 Volume of Rectangular Prisms 6.G.2 M06-S4C4-06

CCSS.MATH.CONTENT.6.EE.A.2.C
Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

CCSS.MATH.CONTENT.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

CCSS.MATH.CONTENT.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Tip of the Day

How often do you tap into the “Problem-Based Interactive Learning” to develop the concept? In my opinion, those pieces are key to helping students develop the conceptual understanding of the lesson. It’s what moves students beyond memorization of steps to understanding the why and the beauty of mathematics!

Learn Zillion

Math Practices

Which two math practices should be in every lesson? The two that I would argue develop mathematical habits of mind: #1) Make sense of problems and persevere in solving them, and #2) Attend to precision.

What does a student do when engaged in the above two math practices?

What does a teacher need to do to set up opportunities for students to engage in the above two math practices? Photo Credit: Steve Rotman via Compfight

Test prep stinks, so actively engage them in review!

I love this Scholastic post that basically says test prep stinks, so actively engage them in their review! Some of their suggestions are:

Some other suggestions for review materials are:

Final thoughts

Remember, just going through worksheet after worksheet of sample tests doesn’t help the students do better on the test. Have them discuss with the whole group or with partners why they chose one answer over another. Have them discuss if it’s reasonable. Engage them!

How do you engage students in review?

Tips , Mathematical Practices, and Ideas about Measurement

Topic 16- Measurement

Original Essential Map dates: March 3-March 7

*Note- many of these topics have been taught previously during double dose time. Rely on assessment data to review these and other topics previously taught in preparation for AIMS.

 16-1 Converting Customary Measures** 6.RP.3.d M06-S4C4-02 16-2 Converting Metric Measures** 6.RP.3.d M06-S4C4-02 16-3 Units of Measure and Precision** 6.RP.3.d M06-S4C4-01 16-4 Relating Customary and Metric Measures** 6.RP.3.d M06-S1C2-02, M06-S1C2-03 16-5 Elapsed Time** 6.RP.3.d M06-S4C4-01, M06-S4C4-02 16-6 Problem Solving: Use Reasoning 6.RP.3.d M06-S5C2-07

** Seen in Double Dose in October-November.

CCSS.MATH.CONTENT.6.RP.A.3.D
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Double Dose Recommendations:

• Gather assessment data for AIMS review.
• Pre-teach the following: Experimental Probability and Predictions (#94-95)

Tech Integration Ideas

In “Creepy Crawlies” (grades 5–8), part of Scholastic’s Math Hunt series, students go on an online fact-finding mission to answer five multiple-choice, math-related questions about bugs and insects. It’s great for integrating science and social studies, math and Internet fluency, and critical thinking.

In addition to challenging students’ reading comprehension skills, “Creepy Crawlies” tests these “math hunting” skills:

• Algebraic Expressions: Order of Operations
• Converting Units of Measurement
• Fractions
• Multiplication/Division

Tip of the Day

If you are departmentalized in 6th grade, I recommend tapping into the Science teacher to help integrate these standards. If you are not departmentalized, why not incorporate a science lesson to give context to when we’d use this math in the real world.

Math Practices

Here’s what the ADE says about 6.MP.7. Look for and make use of structure.

Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent expressions (i.e. 6 + 2x = 2 (3 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality; c=6 by division property of equality). Students compose and decompose two- and three-dimensional figures to solve real world problems involving area and volume.

What are some other resources, ideas, or tips you’d add to this topic?

Tips, Mathematical Practices, and Ideas about Equations and Graphs

Topic 15- Equations and Graphs

Original dates: February 19-27 (Feb. 28 posttest/pretest)

 15-1 Equations with More Than One Operation 6.EE.7 M06-S3C3-02 15-2 Patterns and Equations 6.EE.9 M06-S3C1-01, M06-S3C3-01, M06-S3C3-02 15-3 More Patterns and Equations 6.EE.9 M06-S3C3-01, M06-S3C3-02 15-4 Graphing Equations 6.EE.9 M06-S3C2-01, M06-S4C3-01 15-5 Graphing Equations with More Than One Operation 6.EE.9 M06-S3C3-02, M06-S4C3-01 15-6 Understanding Inequalities 6.EE.8 M06-S1C1-04 15-7 Problem Solving: Act It Out and Use Reasoning 6.EE.5 M06-S3C2-01, M06-S5C2-02

Double Dose Recommendations:

Pre-teach the following: Outcomes and Experiments (#86-91)

Tip of the Day:

Teaching Channel: Daily Assessment with Tiered Exit Cards

Mathematical Practices: #8 Look for and express regularity in repeated reasoning

In grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization. Students connect place value and their prior work with operations to understand algorithms to fluently divide multi-digit numbers and perform all operations with multi-digit decimals. Students informally begin to make connections between covariance, rates, and representations showing the relationships between quantities.

With looking for and expressing regularity in repeated reasoning, students will:

• Look for methods and shortcuts in patterns in repeated calculations
• Evaluate the reasonableness of intermediate results and solutions

Teachers will:

• Provide tasks and problems with patterns (such as multiplying fractions and mixed numbers)

Some question probes could be:

• What is happening in this situation?
• Is there a mathematical rule for?
• What predictions or generalizations can this pattern support?

What resources or tips would you share about solving for equations and graphs?

Tips, Mathematical Practices, and Ideas about Understanding Percent

Topic 14- Understanding Percent

February 7-17 (Feb. 18 posttest/pretest)

 14-1 Understanding Percent 6.RP.3 M06-S1C1-04 14-2 Fractions, Decimals, and Percents 6.RP.3 M06-S1C1-01 14-3 Percents Greater Than 100 and LessThan 1 6.RP.3.c M06-S1C1-01, M06-S1C1-04 14-4 Estimating Percent 6.RP.3 M06-S1C1-01, M06-S1C3-01 14-5 Finding the Percent of a Number 6.RP.3.c M06-S1C3-02, M06-S3C3-03, M06-S5C2-05 14-6 Applying Percents: Finding the Whole 6.RP.3.c M06-S3C3-01 14-7 Problem Solving: Reasonableness 6.RP.3.c M06-S5C2-01
• CCSS.Math.Content.6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• CCSS.Math.Content.6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Double Dose Recommendations

Pre-teach the following:

• Measures of central tendency, 2 days (#76, #78)
• Applying mode, median, mean, 3 days, (#83-85)

Technology Integration

Click here to see the full post by Mr. Avery. If you notice, Mr. Avery asks two guiding questions at the bottom of the post. Have you considered having groups of students respond to the post (or perhaps the whole class compose a response to send them)?

Tip of the Week #1

How would you apply percentages to the real world? Here an idea that would take one class hour, and would engage kids: Trashketball Review for ratios, fractions, decimals, and percentages.

Tip of the Week #2

What would this standard look like? This tip comes from the ADE, with examples and explanations of this standard: Mathematical Practices: #7 Look for and make use of structure

In grade 6, students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent expressions (i.e. 6 + 2x = 2 (3 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c = 12 by subtraction property of equality; c=6 by division property of equality). Students compose and decompose two- and three-dimensional figures to solve real world problems involving area and volume.

What are some specific questions you can ask about math practices during this unit?

Tips, Mathematical Practices, and Ideas about Solving Proportions

Standards

Topic 13 focuses mainly on Ratios and Proportional Relationships (RP) 3, 3a, and 3b:

• CCSS.Math.Content.6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• CCSS.Math.Content.6.RP.A.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
• CCSS.Math.Content.6.RP.A.3b Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

Double Dose Recommendation:

Pre-teach the following:

• Reading Frequency tables and histograms, 2 days (#71-2)
• Stem and Leaf Plots, 3 days (#73-35)

Tip of the Week:

One highlight of my walk-through at FPES a few weeks ago was seeing an awesome cumulative review game in several classes. Math Instructional Coach, Kathleen Donlan, engaged classes in this powerful collaborative group math game. She originally got the idea from a math competition she went to years ago, and has tailored the game to fit the classes with the standards they’ve learned so far.

Math Practices

The Teaching Channel created a great video about including math practices and explicitly in daily instruction.

How do you include the math practices daily?

If you have used Kathleen Donlan’s review game, what insight can you share with others?

Tips, Mathematical Practices, and Ideas about Ratios, Rates, and Proportions

Topic 12- Ratios, Rates, and Proportions

January 21-27 (Jan 28 posttest/pretest)

 Lesson12-1 Understanding Ratios CCSS6.RP.1 Arizona StandardsM06-S1C1-01 12-2 Equal Ratios and Proportions 6.RP.3 M06-S1C1-01 12-3 Understanding Rates and Unit Rates 6.RP.2 M06-S1C1-01, M06-S1C1-03 12-4 Comparing Rates 6.RP.3.b M06-S1C1-01, M06-S1C1-04 12-5 Distance, Rate, and Time 6.EE.9 M06-S1C2-04, M06-S3C3-04 12-6 Problem Solving: Draw a Picture** (2 days) 6.RP.2 M06-S3C2-01

Double Dose Recommendations:

• Pre-teach the following: Mean, Median, Mode, 3 days (#68-#70)
• 12-6  Problem Solving, 2 days

Tip of the Week:

The following tips  come straight from the ADE:

 Standard ADE Explanation and Example(s) CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” CCSS.Math.Content.6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.” CCSS.Math.Content.6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. CCSS.Math.Content.6.RP.A.3b  Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Mathematical Practices:

Attend to Precision — Remind students that ratios can be written as a fraction (3/7), which is the same as 3 to 7, which is the same as 3:7. They all compare the portion of 3 to the whole of 7. Ask students about the math terms they can apply in different situations. — What math terms apply in this situation? or “What math language, definitions, properties can you use to explain ….?

Tips, Mathematical Practices, and Ideas about Two-Dimensional Figures

Topic 11: Properties of Two-Dimensional Figures

Note: The original dates you set for this unit was January 6-16 (Jan 17 posttest/pretest). However, most of you will be starting this unit soon since your pacing is determined by student understanding.

 11-1 Basic Geometric Ideas 6.G.3* 11-2 Measuring and Drawing Angles 6.G.1* M06-S4C4-01 11-3 Step-Up Lesson: Angle Pairs 7.G.5 M06-S4C1-02 11-4 Triangles 6.G.1* 11-5 Quadrilaterals 6.G.1* 11-6 Step-Up Lesson: Circles 7.G.4 11-7 Step-Up Lesson: Transformations and Congruence 8.G.2 M06-S4C2-01, M06-S4C2-02 11-8 Step-Up Lesson: Symmetry 8.G.1 11-9 Problem Solving: Make a Table and Look for a Pattern 6.EE.9 M06-S5C2-07

Double Dose Recommendations:

• Vertex Edge, 4 days, (AZ12, AZ13)

Tip of the Week:

• There is a lot of new vocabulary during this unit; therefore, making connections to vocabulary are extremely important. Click here to view some ideas for integrating technology with vocabulary. It can be used as a center activity in math. For some students, you may even ask the home room teacher if there is any time your student could substitute something (such as the morning wake up review) to spend extra time with vocabulary.
• Kim Sutton also has vocabulary ideas too.

Mathematical Practices:

When students share their thinking  and reasoning about the solutions, they are justifying their solutions (DOK 3),  a foundational critical-reasoning skill. The ability to articulate a clear explanation for a process and critique the reasoning of others is the backbone of Math Practice #3Construct viable arguments and critique the reasoning of others.

Some question stems for MP.3 are:

• What strategy will you try to solve …? How did you decide to try that strategy?
• How did you decide what the problem was asking you to find? (What was unknown?)
• Did you try a strategy that did not work? Why didn’t it work? When would it work?
• How could you demonstrate a counter-example?

Have you thought about having the students solve a problem and capture their thought process on an iPad? Then have students pass the iPad to another group to listen and critique their explanation?

How do you incorporate Mathematical Practices daily?