Math: Side by SideNotice More. Wonder Together.Tools for your side of the table
Make an idea visible. Explore ways to represent numbers, fractions, decimals, and relationships—then talk about what the picture shows.
These bands are starting points, not tests. The overlaps let you revisit an idea or explore what comes next. Choose what fits your child’s thinking.
Count, see parts, and connect quantities to numbers.
Ten-frame
A ten-frame arranges objects in two rows of five. The spaces make groups of five and ten visible.
Look for five and two, or a full ten with three spaces empty. Move seven real objects into another arrangement and discuss what stayed the same.
Part–whole model
A whole amount can be broken into parts and put back together. The lines connect the parts to the whole.
Try four and three, six and one, or seven and zero with objects. All represent the same total; the parts change.
Number rack
A number rack groups beads in fives. Sliding beads together makes parts of a number visible.
Invite five and three or two groups of four. The two beads left over help show that eight needs two more to make ten. Bridges connection: counting and addition models, pages 1–2.
Make tens, build equal groups, and begin exploring equal parts.
Base-ten drawing
Each long rod contains ten little squares. The four separate squares represent four more ones.
Two tens and fourteen ones also make 34. Trading a ten for ten ones changes the representation, not the amount. This supports regrouping.
Array
Rows and columns organize equal groups. Splitting an array connects a new multiplication fact to facts your child already knows.
Two rows have eight counters and the third has four. Together they make twelve. Turn the array to explore four groups of three.
Equal-length strips
The whole matters. Each strip here is the same length, and each is divided into equal parts.
The parts have different sizes. One of two equal parts covers the same length as two of four equal parts. Keep the wholes equal when comparing.
Open number line
Begin at 38 and break 27 into 2, 20, and 5. This scaled drawing shows one possible route; your child can sketch another.
Two reaches 40, twenty reaches 60, and five reaches 65. Compare adding 20 and then 7. Bridges connection: place-value and number-line models, page 3. Try these jumps in Math Meadow →
Connect fractions, decimals, and the size of the whole.
Equivalent fraction strips
Both strips represent one identical whole. Smaller pieces can describe the same shaded length.
The number of equal pieces doubled, and so did the number of shaded pieces. The shaded amount is still three fourths, now named six eighths.
Interactive hundred grid
The full grid is one whole. Each small square is one hundredth; each full row is one tenth. Move the slider to connect the picture, fraction, and decimal.
Five more hundredths are shaded. At 0.40, forty hundredths make four tenths. Ask why 0.4 and 0.40 describe the same amount.
Double number line
The zero and one line up on both number lines. Each fourth covers the same distance as two eighths.
The unit whole and the location stay fixed while the number of equal parts changes. Bridges connection: fractions and decimals, page 7.
Egg-carton model
Here, one whole means all twelve spaces in a carton. Eight filled spaces can be grouped into two of three equal groups of four.
Twelve spaces make the whole set. Each group of four is one third; two groups are filled. Bridges connection: egg cartons, page 7.
Geoboard
Five pegs along each edge enclose four spaces. The entire square is one whole, with sixteen equal square units.
Pegs mark boundaries. Area counts the enclosed square units. Eight shaded squares cover one half of the whole. Bridges connection: geoboards, page 8.
Partial-products area model
Split 12 into 10 and 2, and 14 into 10 and 4. The four regions account for the whole rectangle.
The corner contributes eight square units. Every part matters: 100 + 40 + 20 + 8 = 168. Bridges connection: multi-digit multiplication, page 6.
Make sense of fraction and decimal operations through models and quantities.
Fraction area model
A full recipe needs ¾ cup of oats. Half a batch needs half of that amount. The coral region shows the amount you will use.
Three fourths is six eighths of the cup. Half of those six eighths is three eighths. The factor ½ scales the positive amount down.
Decimal area model
The entire grid is one. Six tenths of its width and four tenths of its height overlap in 24 hundredths.
Four tenths of six tenths covers 24 small squares out of 100, or 0.24. In a price story, 0.6 meters at $0.40 per meter costs $0.24.
Measurement division
You have ¾ cup of rice and a ⅛-cup scoop. Each shaded piece represents one full scoop.
Three fourths is six eighths. Six scoops of one eighth fill three fourths. Division by a small positive amount can produce a count larger than the original number.
Ratio table
Each batch uses ¾ cup of oats. The table keeps batches and cups connected.
Double both quantities. One and a half cups becomes three cups. Bridges connection: ratio tables for fraction and decimal operations, page 10.
Explore rates, percents, and changes that cross zero.
Double number line
Each notebook costs $3 in this example. Matching positions connect the number of notebooks and their total cost.
At a constant $3 per notebook, both quantities scale together. Four notebooks cost $12. This relationship assumes no extra fee or discount.
Signed number line
The temperature starts at −2°C and rises 5 degrees. Position and change are different ideas: −2 is the start, +5 is the move.
Two degrees take us from −2 to zero, and three more take us to 3. Break the change into parts to see why −2 + 5 = 3.
Percent bar
In this example, a $40 item is discounted by 20%. The bar shows five equal parts because 20% is one fifth.
One fifth of $40 is $8: the amount taken off. Four fifths remain, so the price before tax is $32. Always name what the percent refers to.
Connect patterns, graphs, equations, and geometric relationships.
Graph, rule, and context
A rental costs $1 to start and $2 per hour. The rule y = 2x + 1, the graph, and the story describe the same pricing plan.
The graph crosses the vertical axis at 1. Every increase of one hour adds $2, so the slope is 2 dollars per hour. At three hours, the cost is $7.
Right-triangle model
Two perpendicular paths measure 3 meters and 4 meters. Their straight-line shortcut is the hypotenuse of a right triangle.
The Pythagorean relationship gives 3² + 4² = 5², so the diagonal is 5 meters. The right angle is essential; this rule does not apply to every triangle.