Math Investigations

Help with Opening PDF Files

Chapter 19

Part 1: For the problem in the Teacher's Edition, page 518

Provide students with a copy of the Problem Worksheet (PDF file).

Ask the students how they will double the muffin recipe on page 518. Some students may suggest multiplying by 2. If so, tell them that is a good idea but point out that the students have not yet learned to multiply with fractions. Ask if there is another way to do it. In the discussion, bring out that adding the amount of each ingredient to itself would double the amount of that ingredient.

Once students have doubled the recipe, they can see which amounts are whole numbers and which amounts are mixed numbers. Some students will not need to actually double the recipe to answer the question. They can reason that if you double 1/2 you'll get a whole number, so any amount with 1/2 in it will be a whole number when doubled. If you double 1/4 you'll get a fraction, so any amount with 1/4 in it will be a fraction or mixed number when doubled.


whole numbers: butter, flour, blueberries, milk, eggs, baking powder, salt
mixed numbers: sugar

Part 2: Be an Investigator

A good time to do this investigation is after Lesson 5 on subtracting fractions.

Introducing the Investigation

Introduce the investigation by reading aloud the assignment at the top of the first page of the Description of Investigation and Student Report (PDF file), by having one of your students read aloud the assignment, or by having the students read the assignment individually.

Put students in groups of two to four to work on the investigation.

Doing the Investigation

Make sure students understand the table at the top of the page, specifically that they'll need two 1/2-foot boards, one 1/3-foot board, etc.

As you observe students doing the investigation, ask them how long each group of boards that will be cut from a one-foot board is. If the lengths of any group add up to more than one foot, talk to the students about the fact that the boards they begin with are only one-foot long and that, if they have lengths that add up to more than one foot, they won't be able to cut out all those pieces from the board.

Ask students to show their solutions. The solution that uses the least number of one-foot boards is the solution shown below that uses six one-foot boards. Keep showing solutions until students agree on the best solution.


  • Cut six one-foot boards in the following way.
  • 1/4 foot and 3/4 foot
  • 3/8 foot and 5/8 foot
  • 1/8 foot and 7/8 foot
  • 2/3 foot and 1/3 foot
  • 1/4 foot, 1/4 foot, 1/4 foot, and 1/4 foot
  • 1/2 foot and 1/2 foot

Student Report

The letter back to Jay Blue gives students an opportunity to show what they have done.

Houghton Mifflin Math Grade 3