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Starting the year: Logic, Fractions and Similes

Reminding ourselves of our goals:

  1. Higher achievement for all by slowing down the curriculum.
  2. Aiming for visualization and conceptual understanding over memorization.
  3. Providing challenge for fast learners, by going into more depth.
  4. Making math interesting and even fun.

1. Starting With Logic Puzzles

We started the year with several days of logic problems, much like last year  —  see our instructions from last Sept:  WEEK ONE  – A Logic Problem  and WEEK ONE – The Lady and the Tiger ,  and an early classwork/investigation: CW 9_4

Basically, you can use any logic problems – the internet is full of them. Liars and truth-tellers,  brain teasers, riddles. IMPORTANT: Try them yourself, first. They should be easy enough to do, after some head scratching. You’ll run into problems that are too hard or too easy, so caveat emptor. We use a think-pair-share approach to solving these, but other formats would work, too. Continue reading Starting the year: Logic, Fractions and Similes

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Our 5th Grade Math Blog — Version 2.0

Welcome Back!    Here we go – Year Two of our blog. (You can read more about us  here.)

We’ve had a great summer, and approach our second year of blogging with the hope that we can improve on what we’re doing.  Of course, that’s the thing about teaching — you never feel like you’ve ‘solved the puzzle’, like you get it completely right. There are way too many moving pieces: needs, circumstances, learning styles, personalities.  But we all keep trying, improving, tweaking.  There is plenty of data documenting the difficulty of our jobs:  here,  here,  and here.

So we’ve promised ourselves that we’d keep teacher morale in mind: Continue reading Our 5th Grade Math Blog — Version 2.0

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Nearing the end… How are we doing? Self-assessment time for us.

It’s time to look at recent quizzes and evaluate the progress we’ve made this year. Why is it we teachers remember the slips, the failures, the lessons that didn’t work? We’ll try to be honest with ourselves here, and evaluate our outcomes so far for the year. We’re evaluating the three concepts we feel are VITAL, non-negotiable skills for 5th/6th grade.

I. Fractions

We’re still having students draw their fractions as much as possible. Some students lean towards not wanting to draw – to use their algorithmic shortcuts instead. Which is fine! We do approve of algorithms — they’re excellent shortcuts based on centuries of refinement.  Except that we want BOTH! Algorithmic and conceptual knowledge.    So we’ve switched from computation-style problems to word problems in Level 1 on quizzes. Here’s the outcome:  (we decided to use all female pronouns today) Continue reading Nearing the end… How are we doing? Self-assessment time for us.

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Problem-solving beats vocabulary

Sorry we’ve been incommunicado for a while! In the last month, we’ve had one week of outdoor education (camping), one week of spring break(yay!), one week of standardized testing(no comment) and a week of school-wide theme-based learning(fun).

The trick is not to stress about curriculum 🙂 All those other things matter! If we are measuring LEARNING as our main goal, then each of those (except the standardized testing – yuck!) involves lots of learning.

So this week, we had to start by spending a couple days reviewing the unit we started last month – Angles. Here’s what we believe: Continue reading Problem-solving beats vocabulary

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The Case for Withholding Algorithms (For a While!)

In 1998, Tom Carpenter and his colleagues documented grades 1–3 students’ use of invented strategies and standard algorithms. The vast majority of students in the study used some invented strategies. The researchers found that students who used invented strategies before learning standard algorithms showed better understanding of place value and properties of operations than those who learned standard algorithms earlier.

We have seen it over and over: a child is tutored early in memorized arithmetic procedures by a well-meaning parent. Continue reading The Case for Withholding Algorithms (For a While!)

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Decimal Multiplication – a Visual Approach

Our 8th grade teachers tell us that of the most confusing ideas for students is the difference between linear units (the sides of a rectangle) and quadratic units (the area of a rectangle). Students routinely confuse x,  x-squared, and x-cubed, without realizing what each represents.

This concept SHOULD reach back to a conceptual development in 4th and 5th grade: the idea of area versus perimeter units.

Our solution: We use spaghetti to illustrate the linear units on the sides of a rectangle. (Actually, if available, linguine works better because it’s flatter!) Continue reading Decimal Multiplication – a Visual Approach

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Teaching Decimals with Decimal Squares

The most visual, most fun way to learn decimals is with decimal squares.

We use decimal squares we bought from  several years ago. 

They show decimals in tenths, hundredths and thousandths, and the equivalencies are obvious. (see below)

The website has fun interactive games, too, but you need to download the Shockwave app to play them.


Continue reading Teaching Decimals with Decimal Squares

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And Quizzes? How are we doing?

Word doc:   Quiz 3

1. Fractions.

Yay – we’re getting this!  Multiplication plus subtraction with borrowing. Example 3 below is one of the few with a mistake, and its last step is correct, it’s the first step that’s not.

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THIS  problem, however, lead us into the pit of confusion.  Only the last example below is correct, but the others came very close. It tells us what we still need to work on.

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2. (Level 2) Multiplication and (Level 3) Different Bases

Multiplication is going well, too. Our students did a lot of area modeling in 4th grade. Exploding Dots helped many students visibly see what it means to borrow and carry within the place value chart.

About 40% of the students tried the Challenge Questions. These challenge questions were useful in clarifying place value relationships for our fastest students who otherwise would say, “just tell me the steps”!

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