Bar Fractions and Snowflake Fractions

1. Bar Fractions

This activity is adapted from Jo Boaler’s “Mindset Mathematics – Grade 5”, page 106. The only change we made is to use a bar rather than a square. We will use squares for a fraction activity later this week.

Step One (Concrete)

Pass out 5 blocks to each child: 2 Red, 2 Blue and 1 Green(or other color…) We use snap cubes, but any cubes will work. Alternatively, squares of construction paper will work.

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More HW examples

Three more examples of homework.

Our next unit is Patterns, but as we move forward, we continue to review fractions and word problems visually and computationally (Levels 1 and 2) on homework. Some students will need weeks of practice on these two skills visually before they are mastered conceptually. Homework is a great place to allow them to continue solidifying those visual skills. Faster-working students will sail through Levels 1 and 2 in a few minutes and have time to work on challenge problems.

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What About Assessment and Grading? PART THREE

OVERVIEW, Part One: (Tuesday)

1) Changing Unit Planning     (2) Changing Instruction      (3) Changing Homework

PART TWO: (Yesterday)

4. Changing Assessment

PART THREE (today):

5. Changing Grading


5. Changing Grading

Ideally,  grades would not be necessary in a effective math classroom. However, as we move toward that ideal, most schools continue to require that we give grades.

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What Does a Daily Lesson Look Like? Four Days of Activities

October Unit: PATTERNS, especially squares. Homework will continue to review fractions and word problems visually.

What does a daily lesson look like?

1. HW

  • Every other day, students hand in HW, and retrieve and correct the last HW. We mark any mistakes on their HWs, but it is their job to correct them. We put an answer key on the board – it shows work – but they cannot bring a pencil if they come to look at it. They only read it, and return to their seats to correct. For each mistake, they write a “Note to Self” reflection.  “Draw more carefully”, “Only draw unit boxes the same size if they’re the same”, “When adding fractions in 12ths, make all the fraction bars the same size”, for example. HW corrections take 5 minutes or so.
  • We collect their binders about once a week to check corrections.
  • Here are HW #6, and #7 and their keys.
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Can ‘Not Knowing’ be Engaging? Can ‘Figuring it Out’ Be Fun?

Once again, this investigation is not about the answers, it’s about the seeing, thinking, talking, and discovery of patterns. It takes weeks to break students’ misconceptions that math is all about the answers at the back of the book. Children need daily excursions into the territory of “I wonder…”, “What if…”, and “I don’t know; what shall I do?” (best answer – an intrigued shrug!) . As they discover that the patterns are there, on the table (or floor, if that’s where they’re working!) – that they’re there, to be seen, their confidence grows.
We have to be careful not to push, not to squash the delicate tendrils of discovery. We mustn’t judge one student’s work as better than another’s because it is more abstract. Everyone sees what they see. Trust that insights are happening.

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The Collatz Conjecture via Mathpickle

Unbeatable! Combining a Greek myth with an investigation with a reward attached!

We start our investigation of patterns every year with this contest offered by Mathpickle:

We start with any youtube video of the tale of Icarus. Then we show the video from mathpickle (link above). They claim to be offering a $1,000,000 prize to any child who can find a number that does NOT end with one when we follow these 2 rules:

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3 Investigations and 3 Homeworks – Investigating Patterns

INVESTIGATION 1:   Visual Patterns

We spend one lesson investigating patterns we choose from this website:

It is best to start with patterns like #2, #15, #18,  that add a constant number of objects at each step.

Concrete/pictorial students will spend the time they need to build or draw the next few steps in a pattern, and count the new totals at each step. Watch for leaps of intuition when a student gradually begins to see a pattern emerging, and takes the risk of predicting the 10th or the 43rd term.

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