More Word Problems


1. Two More Fun Warm-ups Reviewing Word Problems

We have seen our 8th graders struggle with algebra problems that involve “the number of bills” and the “value of those bills”. They can write x + y = 27 if there are 27 five and 10 dollar bills altogether, but stumble over the value equation:  10x + 5y = 210 when told that the 27 bills add to a value of $210.

So we decided to try to start such distinctions earlier – 5th and 6th grade. Here are 2 Power-Point Warm-Ups that help students begin to make this journey.  As always, use manipualtives (we used Cuisenaire rods and Monopoly money, but any blocks will do) and give them time.

Word Problem- Money


Word Problem – Pears

Photos of the “Money” Problem:


2. Order of Operations Lesson

This is a more traditional lesson, since it represents a convention that mathematicians follow to avoid misunderstandings.

Order of Op

Since we are also 8th grade Algebra teachers, we insist on good algebraic form – what we call the “Sacred V”.

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3. Homework

Here are all the HW assignments from the last unit, plus their Solution Keys:

HW#1  HW#2   HW#3   HW#4   HW#5    HW#6    HW#7    HW#8   HW#9

HW Keys 1-9


‘Mystery’ Warm-Ups, plus: Starting our Unit on Patterns


We found on our second quiz that many students were still struggling with word problems.  (Correct student quiz answers here.)  So we adapted some of  Steve Wyborney’s “Esti-mysteries”  to continue reviewing word problems, and students seem to enjoy these ‘mysteries’ and look forward to them.

Try these engaging warm-ups:

Important:  Take TIME when you show these. We try to slow down the process as much as possible (without totally ruining the tension!), in order to allow more students to spend the time they need thinking through the problems. Number 3 can be used as a “Number Talk’ to see how many ways students can see the problem 7 x 13. Continue reading ‘Mystery’ Warm-Ups, plus: Starting our Unit on Patterns

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”. Continue reading Fractions: Transferring From Blocks to Word Problems

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.

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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:

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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