Materials & Prep
Prepare grid paper or a fraction-model handout, colored pencils, and number lines marked from 0 to 3 in halves, thirds, fourths, fifths, sixths, eighths, and tenths. Copy the partner problem set and the exit ticket. No special room setup is needed. Draw three identical rectangles, each divided into fourths, on the board for the opening.
Opening prediction
Display three identical rectangles, each divided into fourths, with one fourth shaded in each rectangle. Ask students to determine the total shaded amount using what they already know. Students may say 1/4 + 1/4 + 1/4 = 3/4 or count three shaded fourths.
Turn and talk“How did you know the total shaded amount? What stayed the same, and what changed?” Invite students to explain that there are three groups of one-fourth, so the numerator count grows while each piece remains one-fourth.
ConnectExplain that repeated equal groups can be written as multiplication. The three groups of 1/4 can be written as 3 × 1/4. Tell students that today they will use the model first to understand this kind of multiplication, then name the rule that matches the model.
Model with a visual
DisplayDraw a rectangle that is 4 whole units long and 1 unit high. Divide the height into 3 equal strips. Shade 2 of the 3 strips across all 4 units. Explain that each small rectangle has an area of 1/3 square unit, and the shaded region contains 4 groups of 2 shaded rectangles.
Think aloud“The fraction 2/3 tells me that each whole unit contributes 2 shaded pieces, and each piece is one-third. Four whole units give me 4 groups of 2 thirds.” Write:
4 × 2/3 = 8/3 = 2 2/3
ConnectOn a number line, make four equal jumps of 2/3. The jumps land at 2/3, 4/3, 6/3, and 8/3. The final point is 8/3, or 2 2/3. The area model and number line show the same product because both show four groups of 2/3.
ConfrontReturn to the opening. Three groups of 1/4 gave 3/4, and the pieces stayed fourths the whole time. The answer 8/12 would mean multiplying the denominator, but the model has eight pieces, each worth 1/3, not twelve pieces worth 1/12. The denominator stays 3 because the size of each fractional piece has not changed.
Check for understandingAsk students to point to the model and answer, “What does the 8 represent? What does the 3 represent?” Expected response: 8 is the number of one-third pieces, and 3 tells the size of each piece.
Guided practice
ModelDraw a rectangle 3 whole units long and divide its height into 5 equal parts. Shade 2 parts across each unit. Students help count the six shaded fifths. Write:
3 × 2/5 = 6/5 = 1 1/5
Have students draw the same product as three jumps of 2/5 on a number line. Pause after the second jump and ask students to predict where the third jump will land.
ExplainThe model shows why the numerator is multiplied: 3 groups of 2 shaded fifths make 6 fifths. The denominator stays 5 because every piece is still one-fifth. Only after students describe the model, state the rule: To multiply a fraction by a whole number, multiply the whole number by the numerator and keep the denominator the same, then simplify if possible.
Check for understandingStudents solve 2 × 4/7 with a quick sketch. Expected result: 8/7, or 1 1/7. Ask two students to explain why the denominator is 7, not 14.
ScaffoldGive students a three-part organizer labeled “Number of groups,” “Fraction in each group,” and “Total fractional pieces.” Provide the sentence frame, “I have groups of , so I have pieces of size .”
Partner investigation
Have partners complete the problems below. For each problem, they must choose an area model or number line, label the fractional pieces, write a multiplication equation, and explain one step using the organizer. Partner A draws or marks the model first. Partner B checks the number of groups and the denominator, then partners switch roles.
Problem set
- Find 2 × 3/8.
- Find 3 × 5/6.
- Find 4 × 2/9.
- Find 2 × 7/10.
Expected1. 2 × 3/8 = 6/8 = 3/4. 2. 3 × 5/6 = 15/6 = 2 1/2. 3. 4 × 2/9 = 8/9. 4. 2 × 7/10 = 14/10 = 1 2/5.
CirculateAsk, “What does one small region or one jump represent?” and “How does your model show the numerator?” Watch for students changing the denominator or drawing unequal parts. Require students to revise a model if its parts are not equal.
ExtensionStudents compare 5 × 4/5 and 4 × 5/5. They must model both, calculate each, and explain whether the products are equal or different. Expected results: 5 × 4/5 = 20/5 = 4, and 4 × 5/5 = 20/5 = 4. Challenge students to explain why equal products do not mean the two fractions describe the same amount in each group.
Closing check
Exit ticketSolve 3 × 4/7. Show the product with either an area model or a number line. Write one sentence explaining why the denominator remains 7.
Expected3 × 4/7 = 12/7 = 1 5/7. Accept a labeled model with three groups of four-sevenths and a sentence such as, “The denominator stays 7 because each piece is still one-seventh.”
Collect the exit tickets and ask students to compare their model with a partner’s model before turning them in. Use the model, the product, and the explanation as the closing evidence of learning.