Crafting a Teaching Sequence for Extended Intervention

The focus of Monday’s session for grades 3-5 math was how to craft a teaching sequence for extended intervention. Participants worked through the entire process of developing a sequence of module lessons that could be utilized for remedial purposes, filling in learning gaps or supporting enrichment. The day started with examining three types of problems encountered in fourth grade. Participants were then asked to focus on just one of the problems and discuss/think of a sequence of related math problems that would lead to a student being successful at the problem at large. Discussions were centered on the idea of how teaching must be collaborative, not an isolated task. Teachers need to play off of the strengths of their fellow teachers in order to help solidify the vertical foundation being built through the Common Core standards. One highlighted belief was that ”A teacher’s pedagogical content knowledge of the grade levels preceding and following his or her own impacts students’ success daily and is the primary engine necessary to meet the needs of all students.” With that in mind, participants started learning how to build a ladder from a point of strength to the objective.

The process for developing the teaching sequence for intervention is based on a cycle that starts with assessing the student, analyzing, developing a plan, teaching and then re-assessing. After assessing the student (using the module assessment), teachers analyze student work using a mathematical practices protocol that helps identify strengths and weaknesses and also aids in developing questions that can be used to help identify the error or where the “lost” has occurred. In other words, teachers need to find where the crack in the foundation is located and where the last point of success is located. Once identified, teachers can read the corresponding module overview and find where in the overview of module topics and lesson objectives the breakdown occurred. At what lesson or lessons did the crack first appear? Once the crack is identified, teachers can now work on constructing a ladder of complexity, but keeping in mind that traveling up the ladder must be able to be done efficiently. Each rung of the ladder is intended for a 20 minute activity, with the top of the ladder being a task aligned to a final objective. Ladders or intervention plans should not exceed 3 weeks in length.

Strategies for finding the vertical links amongst grade levels included looking at the curriculum map, curriculum overview, foundational standards and the Common Core standards checklists found on EngageNY. Much time and energy was spent on researching within topics and lessons across grade levels to find activities or lessons that help aid in teaching the sequence more deeply. Groups made an illustrated poster to share the sequence and then spent time creating “second” chance assessment questions that allow students to experience and see their growth first hand.

Time and pacing came up as an area of concern. Most agreed that the process presented would work well in aiding AIS instruction. The point was really driven home that teachers need to utilize the strengths of their other grade level teachers on where to find foundational lessons in the modules that directly link to the final objective.

Materials for this session are available here.

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: 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.

Grade 5 Math Module 3: Addition and Subtraction of Fractions

During this morning’s grade 5 math session, participants worked through the Module 3: Addition and Subtraction of Fractions.  The first two lessons (topic A) review grade 4 standards and the concept of equivalent fractions.  Students use paper folding activities to demonstrate equivalent fractions, which helps make the concept very concrete. They use visual modeling via arrays and number lines to help make meaningful connections and the relationships that fractions have to one another.

Topic B is where students encounter fractions with un-like denominators.  They know the language of 1 apple +1 apple, 1 third + 1 third, so when they encounter 1 third +1 half, they know the units are not the same and that they need to make the units the same (common units).  Lots of work with arrays is involved here and there are some struggles, but overall the visual model helps the transition from concrete to abstract.

The group spent a lot of time on a two-step word problem from Lesson 7 that involved subtracting from a whole with uncommon units.  The solution was presented two different ways using visual models (arrays).

For topic C, participants took a look at three student solutions to the problem 3 3/5 – 2 ½.  It was interesting and exciting to see students use unbundling and decomposition that they saw in previous grades with whole numbers, and applying those concepts to their work automatically with fractions.

Topic D focuses on the problem-solving practice and application of the concepts learned. Multi-step problems are tackled with various strategies. The module does a nice job of showing the progression of equivalent fractions, making units pictorially, making units numerically, and then being able to apply knowledge to solve problems that involve addition and subtraction of unlike fractional units.

Living an Elementary ELA Lesson

During Thursday’s afternoon ELA session, teachers took on the role of the student and started with a reading of the non-fiction article “Characteristics of a Multinational Company” in order to come up with the gist of the article. This activity led into another where the session participants could then read and find the gist of another text around which the lesson is based, “The Red Cross.” The participants then used a graphic organizer (3-column Note Catcher) to complete the tasks associated with the Taking Notes Task card protocol. Through close reading of the article, everyone was able to identify key vocabulary terms and find text-based evidence to justify their definitions, just as their students would do. The graphic organizer is a tool that helps students to define a new term/concept based on the information gleaned from the two texts. In this case, the two articles were used to identify what a multinational aid organization is.

Through a Chalk Talk, the participants discussed their own notes based on the articles they read. Then, through the Think Pair Share protocol, ideas were exchanged about the best way to respond when a community is struck by a natural disaster. Participants shared their ideas with the rest of the group in order to explain key ideas from the texts they read. After “removing their student hats,” the teachers discussed the benefits of the protocols and how they are an effective way to address the Speaking and Listening standards. One major benefit was that the protocols helped to grow their thinking through the sharing of mutual ideas and they can offer help to students who need their thinking redirected.

Session participants read the lesson that was the source of the day’s activities and looked for evidence of collaboration, protocols, and conversations. This lesson is a good example of incorporating the Speaking and Listening standards into a Common Core-aligned curriculum. A question was asked about the appropriateness of the protocols for various lessons. Each protocol in the list provided on EngageNY is prefaced with a purpose so that a teacher can decide which protocol to use depending on the goals of the lesson. The session concluded with teachers reflecting on what kinds of classroom management practices need to be in place in order to foster a collaborative environment.

The remainder of the afternoon was spent looking at how protocols can improve literacy skills and be effective in a collaborative classroom environment. Four Corners and Fishbowl were put into practice in this session. The teachers again played the role of the student for the Four Corners activity. Here, the teacher/students had to choose (from 4 options) the most important thing the Red Cross does when communities are struck by natural disaster. Conversations were text-based and gave the teachers an opportunity to see this protocol in action through the eyes of their students. They then participated in the Fishbowl protocol where volunteers discussed the purposes of the Red Cross based on the notes they had taken throughout the “lesson.” The emphasis here is on relevant notes, relevant information, and relevant evidence. Observers of the Fishbowl conversation had to look for evidence of relevance in the conversations along with other qualities including how the participants in the discussion behaved throughout the exercise. Everyone spent the remainder of the session developing a rubric for Speaking and Listening standards.

The activities and texts in these sessions came from Grade 5, Module 4, Unit 3, Lesson 3.

Module Focus: Grade 4 Math Module 3

The morning session for grades 4 and 5 math focused on Grade 4 Module 3: Multi-Digit Multiplication and Division, where students see multiplication and division in action.  The standard algorithm is introduced in grade 4, but it is not a fluency for this grade level.  Multiplication is a fluency in grade 5 and division is a fluency in grade 6, so the intent of the module is to allow for the deep conceptual understanding of the process through the use of modeling techniques.

Participants took a look at division through the use of number bonds, array/area models, and place value/number disk charts.  This exercise provided a great visualization of the process.  Language plays a key role, with the correct use of the terms, “whole,” “quotient,” and “remainder” (how many are left). Phrases like “distributing evenly” and “decomposing” are important in explaining the process.

Teachers need to pick fluency activities that tie into the lesson well.  Participants looked at an example:

How many groups of ____ are in _____? Prove it by counting by ______.

The fluency activity will lead into the concept of “remainders,” an important link to the lesson.  Teachers need to make sure there is a connection with the fluency to the lesson objective.