# Module Focus: Grade 4 Math Modules 6 and 7

The focus of Thursday’s grade 4 presentation for mathematics was on the content of Module 6 and Module 7. Module 6, “Decimal Fractions,” allows students to extend prior knowledge of fractions by seeing decimals as an application of fractions. The progression of the module allows students to see that decimal and whole numbers behave the same way, and that working with decimals just increases their sense of number. Participants were reminded that that even though scaffolding is embedded in the lesson content, teachers may need to provide additional scaffolding measures. It is important that throughout any module, teachers amplify language. Teachers need to use academic language and be clear, effective and consistent. Teachers also need to develop conceptual understanding of the content matter by continually going from the concrete to the pictorial to the abstract. Too many visual representations might be ineffective, so teachers need to be strategic when choosing the best modeling techniques to use for the pictorial. Lastly, teachers need to model strong questioning techniques and demonstrate how to speak and write mathematically. Sentence frames and turn/talk opportunities are some examples of how to accomplish this within a lesson.

Participants started off by looking at the end-of-module assessment and working on question 6. They discussed how they could use the assessment as a planning tool and how it would guide the delivery of the lessons.

Module 6 starts off with students exploring tenths concretely through length, weight and capacity. A scale and pre-filled bags of rice was used to demonstrate how students can decompose 1 unit (kg) into 10 bags or tenths. What does the scale say? 0.1. Other decomposition problems are discussed in this module and the number bond is used. The same methodology is used in earlier grades for bundling tens and working with teen numbers. Teachers saw a strong connection here and were excited, saying “We just need to give this process time.” The overall goal of Topic A is for students to build fluency in writing decimal numbers three ways: as a fraction, as a decimal or in words.

In Topic B, students decompose tenths into 10 equal parts to create hundredths. Students model the meter stick with a tape diagram and quickly learn that tenths make us more efficient when counting hundredths. Sometimes we need to adjust the model depending on our learners; therefore, students not only work with tape diagrams but with area models and number disks to see the equivalence of 1 tenth and 10 hundredths.

Topic C gets students to apply their knowledge gained in the first two topics in order to compare decimals. Students continue using tape diagrams and area models to show their conceptual understanding of the decimal comparisons. Participants did an activity from Lesson 11 that involved cutting out decimal flash cards and ordering the decimal numbers from least to greatest. The decimal numbers were represented in various forms. Participants then needed to plot the decimals on a given number line and determine the best endpoints for the number line.

Topic D introduces the addition of decimals with tenths and hundredths, without using any algorithm. Students become more fluent with their conversions between the two in order to add and use decomposition strategies in the process. Lastly, money amounts as decimal numbers are introduced. Money is used to extend the students’ conceptual understanding of decimals while providing an application of the skills learned. Participants ended the morning session by going back to the end-of-module assessment and discussing what they learned as they were going through the lesson that would need to be reflected in the teaching of the modules.

The afternoon focus for grade 4 was on Module 7 that deals with exploring measurement with multiplication. Since fluency for grade 4 is multi-digit addition and subtraction, core fluency differentiated practice sets are used in this module. Lesson 2 contains the master copies for the 4 practice sets. A great feature of these sets is that each one is broken into 2 parts, with part 2 involving problems that do not involve re-grouping. Both parts, however, have the same answer key, which makes for simple grading.

Module 7 allows students to develop an understanding of the two measurement systems (metric and customary) and allows them to become fluent with converting between larger and smaller units. Great application problems that reinforce the RDW process are found throughout the module and students get to connect their problem solving with mixed units. An interesting approach is taken to time, as conversion is taught with the clock being an unwrapped number line. There is a strong connection here with previous work with number lines.

The module ends with 4 lessons that have year-in-review activities that focus on the area of composite figures, more fluency activities and games designed to solidify vocabulary used throughout the year. The presenters gave ample time throughout the day to work on problems and discuss. This was a very informative presentation that once again demonstrated the progressive nature of the modules and how they are written to build off of the prior knowledge of skills.

The materials for this session are available here:

# Module Focus: Geometry

The focus of the Wednesday morning session for grades 9-10 math was to explore the topics covered in the first two modules of geometry. Congruence is covered in Module 1. The biggest change in geometry with respect to congruence is that two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one on to the other. Rigid motions are first introduced in grade 8 and teachers might want to take a look at Grade 8 Module 2 and Grade 8 Module 3 to see how the properties of rigid motions were explored. Participants discussed how they currently characterize transformations, and most agreed that they associate transformations with a set of rules and they tend to be very coordinate based. Students now need to develop a deeper understanding of transformations and their purpose without the use of the coordinate plane.

Module 1 starts off with 5 lessons on constructions. Students will be performing the same constructions as in the past, but the focus is on not just the figure being constructed, but the steps behind the construction. Students will need precision with their vocabulary, as they will need to be able to communicate clearly the steps behind the construction for all to understand. Focus is on the construction and instruction.

Topic C in Module 1 covers the transformations and rigid motions studied in 8th grade. The progression of intuitive, to the concrete, to formally defining a transformation is developed. Participants took a look at this progression with the concept of reflection in lesson 14 where students explore what they notice about the line of reflection and perpendicular bisectors. They then tie this exploration back to their work in the opening lessons that dealt with constructing the perpendicular bisectors and angle bisectors. Students are then introduced to the formal definition of reflections.

Topic D introduces the concept of congruence through rigid motion. Lesson 22 is the presentation of the proof by rigid motion for the SAS criteria. Students need to know the properties that are preserved with the transformations that are rigid motions (i.e. distance preserving, angle preserving) and need to be able to communicate these properties while writing proofs that involve the use of rigid motions. Once the congruence criteria (i.e. SAS, SSS, HL) have been proven, they then can be used in proofs for congruence as we saw in lesson 26.

Topic G reviews the content of the modules and reinforces the purpose behind the axiomatic system. A math teacher’s story was told: “We have to cover several chapters from the textbook and there are approximately 40 formulas. I may offer you a deal: you will learn just four formulas and I will teach you how to get the rest out of these formulas.” The students gladly agreed.

Module 2 focuses on similarity and right triangle trigonometry. Scale drawings are first introduced in grade 7 and teachers might want to take a look at the content covered in Grade 7 Module 1 for gap purposes. Scale drawings are approached in the geometry module with two methods, the ratio and parallel method. Participants had fun with the parallel method and using the set square to generate parallel lines. After scale drawings are explored, students go on to study the properties of dilations which sets the tone for proving the similarity criteria for triangles (AA, SAS and SSS).

The remainder of Module 2 focuses on right triangle trigonometry. Lessons 16, 21 and 25 set the foundation for the trig functions without officially using the language. These lessons explore the internal relationships within and between similar triangles and how the ratios of corresponding sides can be used to find missing lengths.

The materials for this session are available here:

# Module Focus: Grade 5 Math Module 5

The Grade 5 presentation focused on the material covered in Module 5: Addition and Multiplication with Volume and Area. Participants practiced lots of hands-on activities in order to experience the progression students experience with developing their concept of volume and area. Students first experience calculating volume by building figures and counting unit cubes. Students construct open boxes and calculate volumes by “filling” in the box. They then experience volume pictorially through the use of dot paper and constructing cubes.

There is much discussion in the module about composing and decomposing right rectangular prisms using layers, which helps with students’ conceptual knowledge of what volume actually means. There is no mention of a volume formula in Topic A. Topic B is where the multiplication formula is introduced with the concept of layers. Students also explore the connection between volume in cm and liquid volume in mL. We were able to see the liquid volume increase by 1 mL after the dropping of a cubic centimeter – very cool!

Application problems were presented at this point, such as:

• A small fish tank is filled to the top with water. If the tank measures 15 cm x 10 cm x 10 cm, what is the volume of the water in the tank? Express answer in Liters. What if after a week, water evaporates so that the water level in the tank is 9 cm high? What effect does that have on the volume of the water? How many Liters?
• This is an interesting problem in that students can just take off the “layer” from the original water level, or they can re-calculate with a new height of 9 cm.
• A shed in the shape of a right rectangular prism measures 6 ft. long by 5 ft. wide by 8 ft. high. The owner realizes that he needs 480 cubic feet of storage. Will he achieve this goal if he doubles each dimension? If he wants to keep the height the same, what could the other dimensions be for him to get the volume that he wants?
• This problem lends itself to discussing what happens to volume when you double one dimension, two dimensions, or all three. The “create a sculpture” activity in lessons 8-9 is an opportunity for students to express their creativity, while at the same time apply the concepts and formula of volume to design a sculpture within a given set of parameters. The activity is graded with a rubric used by the students. Participants discussed the value of having students use a rubric. Peer review always holds students more accountable, but the peer review also ties into the Mathematical Practice of critiquing the reasoning of others.

Topic C shifts the focus from volume to calculating the area of rectangles with fractional side lengths.  Once again, this demonstrated an excellent transition from concrete, pictorial to abstract. Students tile a rectangular region using patty paper, then draw the image on white paper (area model), and then use prior knowledge of area (partial products) and the multiplication of fractions to calculate the area. Participants practiced this transition using mystery rectangles. The topic ends with application problems that ask students to decide which process is more efficient and whether they should deal with improper fractions or convert to mixed numbers.  We want them to say “It depends.”

Topic D uses the cutting apart of trapezoids and parallelograms in order to take a look at the properties that exist for each, leading the student towards success in being able to create a hierarchy of quadrilaterals that go from most general to specific. Excellent visual activities were done here with parallelograms constructed by the group so that participants had a wide range of parallelograms. Activities showed the angle relationships that exist within these shapes (consecutive angles supplementary, all four angles add up to 360). Participants looked at diagonals for parallelograms and great questioning techniques were modeled in regards to answering the question, “will the diagonals always be bisected, or are they ever the same?”  Angle measurement was recommended as a fluency activity. An excellent end to an excellent week here at NTI.

# Module Focus: Grade 4 Math Module 5

The Grade 4 Math session on Thursday focused on the Module 5: Fraction Equivalence, Ordering and Operations. Number bonds, tape diagrams, and area models were used consistently throughout the module to strengthen conceptual understanding and develop confidence when working with fractions. This “fractional” confidence allows students to transition to concepts/problems of higher complexity, which was demonstrated as we went through the lessons.

Students start their fraction work off with experiencing problems that involve decomposing fractions using number bonds, similar to the work they did with number bonds and whole numbers in the earlier grades.  How many ways can you represent 5/6 as an addition problem? When answering this problem, students encounter how to express a non-unit fraction as a whole number times a unit fraction. The work here is extended to fractions that are greater than 1, such as decomposing 7/4 = 4/4 + 3/4 .

Fraction equivalence is explored using tape diagrams (paper folding) and area models.  Participants look at an application problem from lesson 5:

A loaf of bread was cut into six equal slices.  Each of the slices was cut in half to make thinner slices for sandwiches.  Mr. Beach used four slices.  His daughter said, “Wow, you used 2/6 of the loaf.” His son said, “No, you used 4/12.” Work with a partner to explain who was correct using a tape diagram.

This problem pulls in all content discussed so far, and solving the problem does not require a fractional algorithm. Fraction equivalence is extended to fraction comparison. Students combine knowledge of benchmark fractions with fraction equivalence to handle comparisons that involve fractions with common numerators, fractions with denominators of related units. The final goal is comparing fractions with denominators of unrelated units.  Topic D shows once again the link of the work done previously with decomposition and composition to the addition and subtraction of fractions with common denominators.  A new visual is added here, the number line with arrows.

Topics E and F add a layer of complexity to what has been learned by extending fractional equivalence and operations to fractions greater than 1. Based on their knowledge, students devise their own strategy for handling problems like 3 3/5 – 4/5. Some might decompose the 4/5 to be 3/5 and 1/5, and then solve the simpler problem of 3 1/5, which is 2 4/5. Others might decompose the 3 as 2 5/5 and now look at the problem 2 8/5 – 4/5, which is 2 4/5. Students practice converting between improper fractions and mixed numbers based on the context of the problem. Never is the traditional algorithm of how to convert a mixed number into an improper fraction discussed.  The module shows that the algorithm is not necessary.  The module ends with a re-visit to repeated addition of fractions as multiplication and shows the connection to the distributive property when solving problems like 2 x 3 1/5.  Visuals are used again here to help make the connection.

The remainder of the session presented educators with a plan on how to make choices with implementing the lessons within the given time frame that remains between now and the assessment. Pacing is a huge area of concern and many teachers are behind the timeline. So how do we adapt the lessons to support successful pacing while bridging gaps in prior knowledge, but not sacrifice the rigor?

A planning protocol was introduced that encourages teachers to look at the lessons further out, not day to day. Reading the module overview and studying the module assessments is the place to start in order to keep the purpose, sequence and delivery fresh in the mind. Next, teachers should read through the lesson and ask what major concept is necessary to successfully complete the exit ticket. Pay attention to the subsequent lesson and examine the exit ticket there. What is the relationship between the two exit tickets and what will be the impact of what gets cut out of the lesson to those two tickets? Teachers also always need to consider the needs of specific students in their classroom.

# Teaching the “Why,” not the “Way”

The focus of the Algebra I Module 4 session was on understanding the method of completing the square when solving quadratic equations. Participants started the session by being asked what they would do in the classroom on day one of teaching this topic. As high school teachers, participants realized that they tend to memorize an algorithm instead of experiencing the “why” behind it; the teachers who have taught completing the square tend to stick to the way to do it, not the why.

Participants looked at lesson 11 and saw an excellent progression of quadratic equations that lead to the need of building a square in order to solve the problem. Connections were presented that would allow students to see the geometric meaning behind the process, focusing on what “complete” really means. This process was extended to quadratics where the lead coefficient was not 1. Prior experience with solving equations and properties would allow students to generate equations where a perfect square could be built. This method would avoid fractions, helpful for some of the students in class. Participants used this method to derive the quadratic formula; very cool indeed! Excellent scaffolding was seen throughout the presentation.

The afternoon was devoted to Topic C, which uses technology to demonstrate and apply previous understandings of the transformations of functions. Lesson 18 presents a great opportunity for the discussion of inverse functions and finding the line of reflection.  Everything was pulled together when looking at lessons 23 and 24 with modeling exercises that explored modeling height over time of projectile objects and the early study of business applications.  The end goal of this module is to be able to have students formulate a general process for solving a quadratic within the context of a given problem.

# Embrace the Challenges, Celebrate the Successes

The Wednesday morning math session started off with a combined grades 6-9 panel discussion about how districts have been implementing the standards and modules.  It clearly is a time to embrace the challenges while celebrating the successes. The greatest challenge seems to be that of time: time to completely grasp and understand the content and presentation of the lessons in the modules prior to teaching them and keeping up with the pacing required.

The issue of time has created a day-to-day routine of re-adjustment for us as educators. Letting go of old methods has been difficult to do, but trusting the work that we and our students are doing because of the depth of understanding now taking place in the classroom is calming some of the anxiety. The concept of closure has changed in that we cannot be so consumed by the final product of the day. We need to focus on and believe in the knowledge, skills, and confidence that are being built daily. Exit tickets do not have to be perfect.

The discussion also focused on how districts can help resistant teachers embrace the change and/or the modules. Some districts have needed to focus on the misconception that the modules are the standards. Teachers must look at the standards first and understand them, and then look to the modules as a tool or resource as one means to accomplish the task of teaching the standards.  Progression workshops where teachers explore the coherency and the instructional strategies for each of the domains is crucial in understanding the standards and has been done in some districts. Another strategy involves using the mathematical strengths of the teachers in the district and allowing them to take the lead for others. Coaching, mentoring, and collaboration have proven essential in these times of change.

# Grade 8 Mathematics: Picking Up Where We Left Off

In the morning session for Grade 8 Mathematics, participants picked up with Lesson 9 in Module 3: Similarity. As a group, participants worked on the exploratory challenge problem that focused on the transitivity of similarity. It was a great exercise that once again demonstrated how the problems in the modules force students to pull from their learning experiences in previous lessons and apply their knowledge to new situations. The embedded ideas here were:

• being able to define what makes triangles similar;
• how to find a scale factor;
• ordering the composition of transformations;
• using the multiplication effect on coordinates; and
• precision of language, i.e., identifying the center of rotations clearly.

In Lesson 10, participants discussed what is meant by an informal argument. In eighth grade, are numeric calculations enough, or do they need to be justified with mathematical statements and evidence?  Participants talked about the importance of building that bridge to high school gand all agreed that students need to be justifying numeric answers with the mathematical facts behind them (i.e., triangle sum).

Another strength of the modules is that most all of the problems force students to use the language of the discipline; if they talk the math, they know the math.  In Lessons 11 and 12, students needed to first determine if there was enough information provided to solve the problem at hand, once again providing an opportunity to critique the reasoning of others, which is one of the mathematical practices.

The modeling problem in Lesson 12 generated a lot of discussion, in particular with interpreting the distance from eye-level “straight down” to the ground.  Pythagorean relationships and where the longest side of a right triangle is found was a helpful guide for the labeling of the “straight down” distance. The vague description in the problem created great classroom dialogue.

Participants ventured on to prove the Pythagorean Theorem using similar triangles, something traditionally seen in high school geometry.  Proof by contradiction is a challenging task and is used to prove the converse.  Teachers might need to walk through some simple “non”-mathematical examples of proof by contradiction first.

Participants spent the remainder of the time looking at the new Module 4 which created a lot of excitement. Connections to similar triangles pop up in Lesson 17 to show y=mx+b. Great ending to a great morning!