Session 4Overview —5th Grade Level II Course
Time / Content / Materials Needed8:30 / Session Starts
- Agenda
- Review Norms
- Overview the Day’s Objectives
8:35 / Explicit Instruction and Inquiry-Based Instruction—When to use which?
Slide 4 - 6 – Introduce the definition and background of explicit instruction highlighting the importance of scaffolding with “I do, we do, you do.” Discuss some of the advantages to teaching explicitly.
Slide 7 & 8 Then have participants watch Anita Archer teach geometry terms in an explicit fashion. Have them record on their T-Chart what characteristics they note about explicit instruction that she demonstrates.
Slides 9-10 Now, in contrast to explicit instruction, let’s discuss inquiry-based learning highlighting the points of it being student-led learning that is facilitated by the teacher. It is project-based and starts with a problem in which students must investigate the solution. Discuss the advantages to this approach to teaching and remind the group to consider when this might be appropriate or inappropriate as an approach to teaching.
Slides 11-12Have them fill in the other side of the T-Chart as they watch a video of Jo Boaler doing an inquiry based lesson. They will note which aspects of inquiry-based instruction that they are seeing as they are watching.
Slide 13--After showing both videos give 5-10 minutes for groups to share their notes and answer the following questions:
Discuss in your group what are the advantages and disadvantages of Explicit and Inquiry-Based Learning.
Look through the Core, and annotate what standards you would you teach explicitly? Inquiry-based? Have a spokesperson who is willing to share your group’s rationale.
Bring the group back together into a guided discussion in which each groups shares their thoughts. Be sure to highlight to the group that students need a solid foundation of skills to be able to do inquiry-based learning. Therefore, if students are learning concepts that are new information it is important for teachers to explicitly teach the concepts so that students are not practicing errors. Once students have this foundation, it is appropriate to start guiding students into inquiry-based learning activities. / Anita Archer Explicit Geometry Video (instructor files movie clips Day 4)
Jo Boaler Inquiry Video (instructor files movie clips Day 4)
Student Journals
10:05 / Slide 14--Time for Math Class…
Slide 15—Activating Prior knowledge related to multiplying and dividing fractions—Table Talk Activity. The goal is to get at how most of us were taught the algorithm for multiplying and dividing fractions with little to no conceptual understanding. The use of strategies was typically non-existence.
5 minutes
Now show the video that is hyperlinked on the slide. This should set the stage for the type of thinking and planning we want to encourage as participants move into the next section of tasks.
Questions to ask:
- What did you see that you are already doing?
- What scaffolding did you see?
- What was the climate for learning? Were the students engaged?
- What are 3 “take aways” that you will use in your own lesson planning?
Slide 17: Take Time to Build the Foundation
This may be a good time to readdress the meaning of the word “of”. Fluency with this term helps students develop a greater understanding of what they are doing when they are multiplying with decimals and/or fractions.
Slide 18--Once again, scaffolding is key to conceptual understanding.
- This is a good place to do a quick review of decimal and fraction place value that includes traditional benchmark values. Ask participants to draw two number lines with 10 equally spaced tick marks. Write 0 on the first mark. Provide the following numbers for participants to place on the number line: 4/8, .75, 2/3, 1/10, ¼. Encourage participants to do a similar activity with their students to build meaning to the size of the parts.
- For number sense, ask participants to explain in words what the following expressions mean: ½ x 19, ½ x ½, and 3 x 2/5. Their answers should reflect an understanding that: ½ x 19 means that you are looking for ½ of a group if there are 19 objects in the group. ½ x ½ means that you are looking for ½ of a group if there is ½ of an object in one whole group. 3 x 2/5 means the total number of objects in 3 groups if there is 2/5 of an object in each group. It is important to note here that when working with fractional or decimal parts, students need a lot of experiences with estimating and proving their answers. They need to be able to articulate the value of their answer as it relates to the quantities described in the problem.
- Students need a strong understanding of what it means to multiply and divide. Ask participants to provide friendly definitions (they can be working definitions) of the two terms. They should do this same activity with the students in their classes and stress the CRA model as it applies to multiplication and division. Students should always be asked to explain what is going on in the problem. For example: Without solving, explain what you are doing if you multiply 0.6 x 8. About how much should your answer be? This is where we develop the understanding that will later lead to the algorithm and determining the placement of the decimal point. Try another one: 8 divided by ¾. With this problem you get right to the heart of the meaning of division of fractions and how to avoid the pitfalls of “invert and multiply”. It is critical to identify the starting quantity and what is happening to it.
Slide 20: Connecting to Whole Numbers
Giving an example of how whole numbers and decimals connect to fractions.
Slide 21: Provide each participant with the handout Multiplying Menace – A Closer Look. Read the story aloud or if possible project the pages via a document camera. They will be looking for the expressions and/or equations that are presented on the pages listed. Participants will complete the graphic organizer as you read through the story. / Student Journals
Number Lines(1 copy per participant double sided)
Fraction Circle Manipulatives
Multyiplying Menace (instructor needs copy of book)
Multiplying Menace—A Closer Look (1 copy per participant)
Modeling your Thinking Video (instructor files Day 4 movie clips)
12:00 / Lunch
12:45 / Slide 22 – We begin our discussion of division of and with fractions by looking at the decimal representation or fractions with a denominator of 10. Students can readily grasp dividing $2.40 into 4 equal groups when they are given the manipulatives to model the problem. When working with monetary quantities, teachers can provide students with dollars and dimes and ask a question regarding equally sharing the money between 4 people.
From here we begin to develop understanding of the dividend, the divisor, and the possible quotient. Scaffolding helps to bring “reasonableness” to student thinking.
Slide 23-- Avoiding “Invert and Multiply” is a critical segment and one that is most often misunderstood by teachers and students alike.
Slide 24---Video to demonstrate scaffolding and connection of whole number division to dividing fractions.
Slide 25– Participants should note that in Problem 2, there isn’t an easy equal share when looking at dividing the fractional parts. Each of the fourths in Problem 2 needs to be subdivided into three equal parts (or twelfths).
Have them TRY IT: Try modeling and/or representing each problem. What mathematical understanding must students have to solve Problem 2? How would you scaffold this increase in difficulty?
Slide 26—Similar activity to 25
Slide 27—A few more problems…Model or represent each of these problems. What mathematical understandings must students have before they can solve each level of problem? Brainstorm with your group, how to scaffold the movement to each level.
Slide 28--Once students are fluent in modeling and representing division by and with fractions, you can move towards the standard algorithms.
- Common Denominator Algorithm
a. Represent this problem using an area model and then write it with common denominators.
b. Make sets of 3/6 from the 10/6. How many sets did you make?
Slide 29—Finally, Invert and Multiply. This slide takes you step by step through why the algorithm actually works. Help participants to make the connection between what was done conceptually and how it gets represented in this algorithm.
Slide 30—Strategy Graphic Organizer--Assignment Outline
Half of groups are doing division, half are doing multiplication
Basic idea:
- Come up with a “real world” problem for both decimal fractions and for whole numbers
- Present the problem to the class with the instructions that the class needs to solve the problem without the use of an algorithm; they need to draw a picture of the problem or if possible, manipulatives.
If Time or for Fast Finishers – provide a copy of Create Break for Multiplication and Division of Fractions. Encourage participants to create a dice game that provides an opportunity for students to practice multiplication and/or division of fractions. / Fraction Manipulatives
Strategy Graphic Organizer (2 per participant—one for multiplication and one for division)
Dividing Fractions Video (instructor files Day 4 movie clips)
Create a Break Multiplication and Division Fractions (1 copy per participant)
Create Break for Multiplication and Division of Fractions
(1 per group)
2:00 / Afternoon Break
2:15 / Putting It Altogether
Slide 31--The big idea of this closing activity is to wrap all the things they have learned together this week into one culminating task. Knowing that each audience will be different, please adjust this task accordingly. For some groups, they will be able to handle this task and for others you may need to scaffold back a bit.
The ultimate outcome is for them to take what they have learned over the last 3 ½ days and create a plan for meeting the demands of one of the standards while applying the core, formative assessment, DOK, explicit instruction, inquiry instruction, C-R-A model, etc.
Slide 32—A refresher of the many topics we covered over the last four days.
Slide 33—Learning Task—Application
Ask participants to select a core standard(s) that they would like to take some time fleshing out for their instruction next year. Have them consider what their learning objectives for the standard they identify are. Then ask them to use their district curriculum map (if they have one), their district adopted materials (if they have them), and their notes from this week to support them in developing a unit of study that includes: Formative Assessment, learning tasks at different DOK levels, explicit instructions, inquiry-based learning, student self-assessment, the C-R-A model, etc.
Participants may want to consider working together in small groups of 2-4 people to work on this task. They may work alone if they so choose as well. As instructors, monitor their work and provide feedback as necessary. This is their time to pull together what they have learned and turn it into an instructional plan they can use.
Depending on your group, you could leave 10 minutes near the end to have them share out what they came up with. Hopefully, they will be willing to share their work with each other in an electronic format so everyone can benefit from their ideas, but if not that is okay / District Curriculum Materials (each participant should bring a manual or topic folder for a standard they would like to work on)
3:30 / Session Ends—Closing
Close the session by having participants respond to the Utah Standards Academy Evaluation. They will be sent a link via their email. This survey is part of them earning their credit/points.