Fractions: Transferring From Blocks to Word Problems

Word Problems – YAY!
Our intensive use of Cuisenaire rods came to fruition now as we attempt to transfer fraction visualization to word problems. Here are the 2 lessons we spent on this so far:

CW Frac to WP1  and  CW Frac to WP2

We used a format called “Builders and Scribes”.

  • Group students in pairs. We then sent each pair to the white boards (we have very long white boards), but they could work at mini-white boards at their desks, too.
  • One student starts as the “Builder”, the other as the “Scribe”. The builder uses Cuisenaire rods, laid out on a mini-white IMG_4518board or clipboard.  (no pencil/pen for the builder). The scribe uses a white board pen (no blocks for the scribe) and records what the builder has built. The builder stands next to their scribe and they set to work. Of course, there’s a lot of collaboration and role-sharing at both ends of this; that’s absolutely fine 🙂
  • There are hints for each word problem, but wait as long as possible before showing them. Our first goal is to MAXIMIZE THINKING TIME, not necessarily to get right answers quickly. However, we also want to avoid completely losing the students, too. Perhaps there’s a Golden Confusion Level 🙂  Let them struggle and think and try, then give a little boost up.pit.jpg
  • After each problem, ask for a little feedback. “What was the hardest wall to get over?”
  • Then the partners switch roles for the next problem.



HW#6  and  HW#6_answ_key

HW#7 and  HW#7 key

HW#8  and HW8_key

Quiz 1  and  Quiz1_key

Here’s a Cuisenaire riddle/puzzle for CW, too, if there’s time: CW 10_2 Cuis_puz


Why Withhold the Fraction Algorithm?

According to this article by Jo Boaler — professor of mathematics education at Stanford and co-founder of  — math memorizers scored poorly on the international PISA test, and the U.S. has more memorizers than most other countries in the world. The highest achieving students internationally were those who thought of math as a set of connected, big ideas.

Here’s what we see:

1. A visual approach to fractions gives students better number sense, and better access to word problems.

When we require drawing, every problem becomes a word problem.   In the problem below, all students recognized that 1/2 is 6 out of 12, visually. This is a “12-peak Toblerone”, so a total of 17 twelfths (by simply counting!) . Then this student imagined moving one 12th from the top row to make the 2nd row equal to one, leaving 5/12 on top. This shows number sense! Our students can do fraction addition and subtraction mentally. More importantly, visualization helps facilitate the transfer to word problems, as below.

Screen Shot 2019-10-06 at 2.16.35 PM.png

Egyptian fractions: We spent a few days answering word problems by building fractions with Cuisenaire rods. Here, for example, is a TWELVE-WIDE wall:index.pngOne fourth —  the light green rod — is called one fourth because four of them fit in a whole.  The purple rod is called one third because 3 of them fit, the red is 1/6, etc.

This student had no trouble finding a way to make 11/12 with Egyptian fractions:

Screen Shot 2019-10-06 at 2.28.41 PM.png

After long exposure to physical representations, word problems become easier. This problem, for example, would be difficult to do with algorithms.


How about this problem:  Erin and Kana went shopping for groceries. Each of them had an equal amount of money at first. Then Erin spent $80 and Kana spent $128. After that Kana had 4/7 of what Erin had left. How much money did Erin have left after shopping? Solve by drawing a fraction model.

This is very difficult to do without algebra. Try it yourself before looking at the answer here. Once you see the solution, it’ll make sense, and all of this will transfer to stronger algebra students in 3 years.

2. A visual approach to math is the ONLY approach that works for some students.

In the past, visual learners struggled with the algorithmic manner in which math was taught. (Challenge: randomly survey a couple dozen adults – we predict almost 1/3 of them will say they were ‘never very good at math’)

However, in the past, there were good middle class jobs available to high school graduates – jobs that are now disappearing. It is our duty to make math accessible to ALL students.
The good news is that requiring visualization of math also benefits the innately abstract math learners. Visualization skills helps students in Chemistry, Physics, Trigonometry, and other STEM subjects these students gravitate towards. Here’s  an article about visualization in physics.







Homework – How Much? How Hard?

Ah… Homework

Raise your hand if you remember hating homework as a child …   Raise your hand if you have children and hate it when they have homework … Raise your hand if, as a teacher, you have ever received homework submitted with teardrops on it 🙁     If you haven’t raised your hand yet, you were born under a lucky star, or you have a faulty memory. face-with-tears-of-joy_1f602.png

Untitled.pngOn the plus side, homework

  • offers a chance for students to independently consolidate skills they learned in a group setting
  • builds skills of responsibility and time management

Continue reading Homework – How Much? How Hard?

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

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

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.

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