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Illustrating Fraction and Whole Number Products with Tape Models
Students will be able to use tape diagrams to create equivalent expressions involving whole numbers, fractions, and multiplication.
- In front of your class, draw a rectangle divided horizontally in half. The upper half should be divided into two equal parts and the lower half into six equal parts.
- Have students think, pair, and share a possible math expression the illustration might represent. Share out student ideas to the whole class.
- Tell your students the illustration can serve as a powerful math tool called a tape diagram, a diagram that expresses equality between a lower level amount, and an upper level amount.
Explicit Instruction/Teacher modeling(10 minutes)
- Using the introductory illustration, draw two dots within each of the six equal parts and six dots within each of the two equal parts.
- Show your students how the diagram represents (6 x ____ ) = (2 x ____ ). Have your students tell a neighbor what amounts they think might go in the missing areas, before you share the missing parts.
- Reveal the equation to read [(6 x 2/12) = (2 x 6/12)]. Tell students this tape diagram shows the two different expressions are equivalent, or they have the same value. The equal sign between the two expressions creates an equation.
- Have students give a signal to show if they suspected this to be the answer. Solve both sides of the equation to show that the expressions are equivalent because they have equal values.
- Label the illustration as you point out the following details to your students:
- both levels are the same length, signifying equality,
- the total number of items on each level show denominator amounts,
- groupings represent numerator amounts,
- the number of groups represent whole number factors.
Guided Practice(5 minutes)
- Draw another rectangle with a horizontal line that splits the rectangle into two equal levels. The upper half should not be divided and the lower half should be divided into three equal parts. Place three dots in the top level and one dot per group for the lower level.
- Show your students the equation (1 x____) = (3 x____) and ask them tell a neighbor what amounts are missing from the equation. Allow for student responses and confirm, (1 x 3/3) = (3 x ⅓), while referring to the guided practice diagram.
- Answer any clarifying questions and solve the equations to show the two expressions are equivalent to each other.
Independent working time(15 minutes)
- Post the following exercises on the board, by drawing:
- a tape diagram model using dots illustrating (4 groups of 4) on top = (2 groups of 8) on bottom with the equation (4 x____) = (2 x____),
- a tape diagram model using dots illustrating (9 groups of two) on top = (6 groups of three) on bottom with no equation,
- a tape diagram model using dots illustrating (5 groups of four) on top = (10 groups of two) on bottom with no equation.
- Have students provide complete equations for each tape diagram.
- Provide multiple tape diagram models with equations missing only one of these components:
- a whole number factor
- a fraction factor
- both whole number factors
- both fraction factors
- Have students reference the opening example illustration to promote independent understanding.
- Have students draft their own tape models and accompanying equations including fractions and whole numbers as factors.
- Confident students may use single-digit numbers instead of dots in their tape model groupings.
- Google sheets is an excellent resource for drafting two-layer tape models using the merge cell and outline cell functions. Explore the application with students to draft their own tape diagram models using dots.
- Show your students two different cards:
- one with a tape diagram and two possible equations,
- one with an equation and two possible tape diagrams.
- Have your class tell you the matching options with explanations for each.
Review and closing(10 minutes)
- Review and confirm answers for the exercise problems, using the opening diagram model as a reference.